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enBig Bang Reality Check.
http://teramari.us/node/129
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>I was reflecting on some recent ideas about the "human centric" understanding of our universe. Our minds and bodies are naturally designed to work within a certain size of objects, quantities, spaces and within human perception. And for everything outside that space we scale things to fit our understanding.</p>
<p>For example we zoom in on and normalize small objects on computer screens until they fit inside our peripheral vision, just like we do with objects normally within our purview. And very large objects we zoom out. For very large counts we have developed methods to organize, quantize and analyze data. We often find ways to visualize or auralize data because we also remap all the things we can't develop (e.g. radio waves) into the visual or audible spectrum.</p>
<p>So I was thinking that if we fit our entire understanding of the universe into our human-centric perception then our understanding of the big bang must also fit into that perception. And an important thing to realize is that there are limitations to our understanding of the universe because there are very clear limits to the things that any human being will ever understand (btw studies on whether or not aliens are also bound by these rules are tbd). So if all these things are true then what we know about the big bang, the end of things and the end of the universe are actually imposed by us human beings on the universe.</p>
<p>And further more if everything is just a concept, contrived by people to help us understand the universe, and that concepts are not real, and that we will never really know the <em>true nature</em> of the universe, then things like the age of the universe, the accelerating universe and all the other unpleasant conclusions may not matter and may not even be true.</p>
<p>But hold on, I'm far from a science denier. I do think that the scientific conclusions about the universe are, at the very least, the best understanding we've come to as of yet and of course that science itself will only be able change these conclusions. But it does make me question the nature of reality and to what extent do we really know the universe and can it be proven that we know something about the universe?</p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/10">Thought & Mind</a></div><div class="field-item odd"><a href="/taxonomy/term/4">Philosophy</a></div><div class="field-item even"><a href="/taxonomy/term/5">Metaphysics & Ontology</a></div></div></div>Sat, 12 Aug 2017 18:46:38 +0000will129 at http://teramari.ushttp://teramari.us/node/129#commentsCalculus: tbd (Part 5 of x)
http://teramari.us/node/128
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>.</p>
</div></div></div>Sat, 12 Aug 2017 18:24:14 +0000will128 at http://teramari.ushttp://teramari.us/node/128#commentsCalculus: Integrals (Part 4 of x)
http://teramari.us/node/127
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Preface</strong><br />
This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.</p>
<p><strong>Introduction</strong><br />
This article will cover integrals.</p>
<p>There are many calculus references, the one I like to use is <a href="https://www.amazon.com/Calculus-Ron-Larson/dp/1285057090/">Calculus by Larson, Edwards and Hostetler</a>.</p>
<p>This also assumes you are familiar with Python or can stumble your way through it.</p>
<p><strong>Concepts</strong><br />
None</p>
<p><strong>Definition of an Integral</strong><br />
The integral of <em>f</em> on the closed interval <em>[a,b]</em> is given by<br /><em>lim<sub>Δ->0</sub> Σ<sup>n</sup><sub>i=1</sub> f(c<sub>i</sub>) Δx<sub>i</sub> = ∫<sub>a</sub><sup>b</sup> f(x)dx = F(b)-F(a).</em></p>
<p>Therefore integrability implies continuity. But continuity does not imply integrability.</p>
<p><strong>Properties of Integrals</strong></p>
<table><tr><td>General Form</td>
<td><em>∫ u dx</em></td>
</tr><tr><td>Constant</td>
<td><em>∫ c dx = cx + C</em></td>
</tr><tr><td>Sum/Difference</td>
<td><em>∫ [u +/- v] dx = ∫ u dx +/- ∫v dx + C</em></td>
</tr><tr><td>Chain</td>
<td><em>∫ f(u) du = F(u) + C</em></td>
</tr><tr><td>General Power</td>
<td><em>∫ u<sup>n</sup> du = u<sup>n+1</sup> / (n+1) + C</em></td>
</tr><tr><td>By Parts</td>
<td><em>∫(u dv) = uv - ∫(v du) + C</em></td>
</tr></table><p>Where u and v are a functions, c and n are constants.</p>
<p><strong>Integral Table</strong><br /><a href="http://integral-table.com/">Here.</a></p>
<p><strong>Integrals of Vector Functions</strong><br />
If <strong>r</strong> is vector-valued function <strong>r(t)</strong> = f(t)<strong>i</strong> + g(t)<strong>j</strong> + h(t)<strong>k</strong>, then</p>
<p>∫ <strong>r(t)</strong> = [∫ f(t) dt] <strong>i</strong> + [∫ g(t) dt] <strong>j</strong> + [∫ h(t) dt] <strong>k</strong></p>
<p><strong>Average Value of a Function</strong><br />
The average value of a function is given by the following formula:</p>
<p><em>[1/(b-a)] ∫<sub>a</sub><sup>b</sup> f(x)dx</em></p>
<p><strong>Initial Conditions</strong></p>
<ol><li>Integrate the function</li>
<li>Substitute initial condition and solve for C.</li>
<li>If integrating again return to first step and repeat until finish.</li>
</ol><p>An example of this is integrating the acceleration constant to determine velocity. You can substitute known initial position values to calculate the constants. Integrate again to get the position function. Then you have specific solutions for your problem.</p>
<p><strong>Multiple Integration</strong><br />
The rules for integrating more than once are identical to a single integral, you just have to observe two things:</p>
<ol><li>The order of operation is inside out. The innermost integral corresponds to the innermost integration variable (e.g. dx, dy or dz). You must have the same number of integrals and integration variables.</li>
<li>Each integration is <em>partial</em>. That is you hold all other variables as <em>constants</em> when integrating.</li>
</ol><p><strong>Surface Area: In the Plane</strong><br />
For a in a plane (flat surface) and bound on four sides by equations f(x), g(x), m(x) and n(x), the area is given by:</p>
<p>A = ∫<sub>f(x)</sub><sup>g(x)</sup> ∫<sub>n(x)</sub><sup>m(x)</sup> dy dx</p>
<p><strong>Surface Area: Multiple Integrals</strong><br />
If f(x,y) and its first partial derivatives are continuous on the closed region R, then the area of the surface S over R is:</p>
<p>S = ∫<sub>S</sub> ∫ f(x,y,z) dS = ∫<sub>R</sub> ∫ f(x,y,g(x,y)) dS</p>
<p>where<br />
dS = sqrt[1+ f<sub>x</sub>(x,y)<sup>2</sup>+f<sub>y</sub>(x,y)<sup>2</sup>] dA</p>
<p>If <strong>r(u,v)</strong> = x(u,v)<strong>i</strong> + y(u,v)<strong>j</strong> + z(u,v)<strong>k</strong> and <em>D</em> be a region in (u,v), then the surface area surface S:</p>
<p>S = ∫<sub>S</sub>∫ f(x,y,z) dS = ∫<sub>D</sub>∫ f( x(u,v), y(u,v), z(u,v) ) dS</p>
<p>where<br />
dS = abs(<strong>r<sub>u</sub></strong> cross <strong>r<sub>v</sub></strong>) dA</p>
<p>where<br /><strong>r<sub>u</sub></strong> = dx/du (partial derivative)<br /><strong>r<sub>v</sub></strong> = dy/du (partial derivative)</p>
<p>If <strong>r(u,v)</strong> = x<strong>i</strong> + y<strong>j</strong> + f(x,y)<strong>k</strong> you will derive the same equation as above.</p>
<p>If <strong>r(u,v)</strong> = u<strong>i</strong> + f(u)*cos v<strong>j</strong> + f(u)*sin v<strong>k</strong> you will derive the same equation as surface of revolution.</p>
<p>S = ∫<sub>S</sub>∫ <strong>F ⋅ N</strong> dS = <strong>F ⋅ (r<sub>u</sub> cross r<sub>v</sub>)</strong> dA = ∫R∫ <strong>F ⋅ [-g<sub>x</sub>(x,y)i-g<sub>y</sub>(x,y)j + k]</strong>dA</p>
<p><strong>Surface Area: Surface of Revolution</strong><br />
If a smooth curve C given by y=f(x), then the area S of the surface is given by</p>
<p>S = 2π∫<sup>b</sup><sub>a</sub>f(x) sqrt(1+(f'(x))<sup>2</sup>) dx (revolved by x axis)</p>
<p>And for x=g(y)<br />
S = 2π∫<sup>b</sup><sub>a</sub>g(y) sqrt(1+(g'(y))<sup>2</sup>) dy (revolved by y axis)</p>
<p><strong>Volume: Disc Method</strong><br />
Find the volume of a function rotated about an axis and perpendicular to the axis of revolution.</p>
<p><em>Horizontal Axis of Revolution</em><br />
V = π ∫<sup>b</sup><sub>a</sub> f(x)<sup>2</sup>dx</p>
<p><em>Vertical Axis of Revolution</em><br />
V = π ∫<sup>d</sup><sub>c</sub> f(y)<sup>2</sup>dy</p>
<p><strong>Volume: Shell Method</strong><br />
Find the volume of a function rotated about an axis and parallel to the axis of revolution.</p>
<p><em>Horizontal Axis of Revolution</em><br />
V = 2π∫<sup>b</sup><sub>a</sub> p(y)h(y)dy</p>
<p><em>Vertical Axis of Revolution</em><br />
V = 2π∫<sup>d</sup><sub>c</sub> p(x)h(x)dx</p>
<p><strong>Volume: Multiple Integrals</strong><br />
If f is defined on a closed, bounded region R, then the double integral of f over R is:</p>
<p>∫<sub>R</sub> ∫ f(x,y) dA, where dA=dx dy.</p>
<p>If f(x,y,z) is continuous over a bounded solid region Q, then the triple integral of f over Q is:</p>
<p>∫ ∫<sub>Q</sub> ∫ f(x,y,z) dV, where dV=dx dy dz.</p>
<p><strong>Length: Arc Length</strong><br />
Let the function given by y=f(x) be a continuous function on interval [a,b]. The arc length s of f between a and b is:</p>
<p>s = ∫<sup>d</sup><sub>c</sub> sqrt[1+f'(x)<sup>2</sup>]dx</p>
<p>For a vector valued function <strong>r(t)</strong><br />
s = ∫<sup>b</sup><sub>a</sub> sqrt[x'(t)<sup>2</sup>+y'(t)<sup>2</sup>+z'(t)<sup>2</sup>] dt = ∫<sup>b</sup><sub>a</sub> abs(<strong>r'(t)</strong>) dt</p>
<p>For a linear (straight line through origin) this function reduces to the standard m=(y2-y1)/(x2-x1) function.</p>
<p><strong>Length: Line Integral</strong><br />
This is for weighted functions. Evaluating the line integral for an unweighted function (f(x,y,z)=1) gives you the arc length above.</p>
<p>Let f be continuous in a region containing a smooth curve C. If C is given by <strong>r(t)</strong> = x(t)<strong>i</strong> + y(t)<strong>j</strong> + z(t)<strong>k</strong> then</p>
<p>s = ∫<sub>C</sub> f(x,y,z) ds = ∫<sup>b</sup><sub>a</sub> f(x(t), y(t), z(t)) ds</p>
<p>where ds = abs(<strong>r'(t)</strong>) dt = sqrt[x'(t)<sup>2</sup>+y'(t)<sup>2</sup>+z'(t)<sup>2</sup>] dt</p>
<p>∫<sub>C</sub><strong>F ⋅ dr</strong> = ∫<sub>C</sub><strong>F ⋅ T</strong> ds = ∫<sup>b</sup><sub>a</sub><strong>F</strong>(x(t), y(t), z(t)) ⋅ <strong>r'(t)</strong> dt</p>
<p>∫<sub>C</sub> 1 ds = ∫<sup>b</sup><sub>a</sub> abs(<strong>r'(t)</strong>) dt</p>
<p><strong>Special Relationships</strong><br /><em>Fundamental Theorem of Line Integrals</em><br />
If <strong>F(x,y)</strong>=M<strong>i</strong>+N<strong>j</strong> then</p>
<p>∫<sub>C</sub><strong>F ⋅ dr</strong> = ∫<sub>C</sub> ∇f ⋅ <strong>dr</strong> = f(x(b),y(b)) - f(x(a),y(a))</p>
<p>where f is a potential function of <strong>F</strong>. In other words for conservative functions the line integral is simply the difference between the potential function of the endpoints and is INDEPENDENT OF PATH.</p>
<p><em>Divergence Theorem</em><br />
Let Q be a solid region bounded by a closed surface S oriented by a unit normal vector directed outward from Q. If F is a vector field whose component functions have continuous partial derivatives in Q then</p>
<p>∫<sub>S</sub>∫ <strong>F ⋅ N</strong> dS = ∫∫<sub>Q</sub>∫ div <strong>F</strong> dV</p>
<p>Both integrals describe total flux. The one on the left evaluates flux across the surface and the one on the right describes flux through the volume. <strong>N</strong> is the unit normal vector.</p>
<p><em>Stokes's Theorem</em><br />
Let S be an oriented surface with unit normal vector <strong>N</strong>, bounded by a piecewise smooth simple closed curve C. If <strong>F</strong> is a vector field whose component functions have continuous partial derivatives on an open region containing S and C then</p>
<p>∫<sub>C</sub><strong>F ⋅ dr</strong> = ∫<sub>S</sub>∫ <strong>(curl F) ⋅ N</strong> dS</p>
<p>This theorem calculates the circulation of the equation about a point.</p>
<p>∫<sub>C</sub><strong>F ⋅ dr</strong> f(x)dx = ∫<sub>C</sub> ∇f ⋅ <strong>dr</strong> = f(x(b),y(b))-f(x(a),y(a)) where <strong>F(x,y)</strong> = ∇ f(x,y)</p>
<p><strong>Relationship between derivatives and integrals</strong></p>
<ol><li>Derivatives look at the tangent line and is concerned with the relationship of dy/dx (division); while the Integral looks at the area and is concerned with the relationship of dy*dx (multiplication). (As a matter of fact, for a straight line the derivative and integral reduce to division and multiplication, respectively.)</li>
<ol><li>∫<sub>a</sub><sup>b</sup> f(x)dx = F(b)-F(a)</li>
<li>(d/dx) ∫<sup>x</sup><sub>a</sub> [f(t)dt] = f(x)</li>
</ol><li>The integral is the antiderivative of a function. One operation is the inverse of the other.</li>
<li>If your function is conservative then the integral of the line is <em>path independent</em>. The line integral of a closed curve is always <em>zero</em>.
</li></ol></div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/13">Hardware & Design</a></div></div></div>Sun, 02 Jul 2017 19:24:51 +0000will127 at http://teramari.ushttp://teramari.us/node/127#commentsCalculus: Derivatives (Part 3 of x)
http://teramari.us/node/126
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Preface</strong><br />
This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.</p>
<p><strong>Introduction</strong><br />
This article will cover derivatives.</p>
<p>There are many calculus references, the one I like to use is <a href="https://www.amazon.com/Calculus-Ron-Larson/dp/1285057090/">Calculus by Larson, Edwards and Hostetler</a>.</p>
<p>This also assumes you are familiar with Python or can stumble your way through it.</p>
<p><strong>Concepts</strong><br />
None</p>
<p><strong>Definition of a Derivative</strong><br /><em>For a function of one variable</em>: the derivative of <em>f(x)</em> at <em>x</em> is given by<br /><em>f'(x) = [f(x+dx)-f(x)]/[dx] = m, the slope of the line<br />
provided the limit exists.</em></p>
<p><em>For a function of more than one variable, partial</em>: the partial derivative of <em>f(x,y,z)</em> at <em>x</em> is given by<br /><em>∂/∂x = f<sub>x</sub>(x,y,z) = [f(x+dx,y,z)-f(x,y,z)]/[dx]<br />
provided the limit exists.</em></p>
<p><em>Differentiability implies continuity. But continuity does not imply differentiability.</em></p>
<p><strong>Properties of Derivatives</strong></p>
<table><tr><td>General Form</td>
<td><em>u dx</em></td>
</tr><tr><td>Constant</td>
<td><em>c dx = 0</em></td>
</tr><tr><td>Constant Multiple</td>
<td><em>cu dx = cu'</em></td>
</tr><tr><td>Chain</td>
<td><em>f(u) dx = f'(u)u'</em></td>
</tr><tr><td>General Power</td>
<td><em>u<sup>n</sup> dx = nu<sup>n-1</sup>u'</em></td>
</tr><tr><td>Sum/Difference</td>
<td><em>[u +/- v] dx = u' +/- v'</em></td>
</tr><tr><td>Product</td>
<td><em>uv dx = uv' + vu'</em></td>
</tr><tr><td>Quotient</td>
<td><em>(u/v) dx = (vu' - uv')/v<sup>2</sup></em></td>
</tr></table><p>Where u and v are a functions, c and n are constants.</p>
<p><strong>Derivative Table</strong><br /><a href="http://www.math.com/tables/derivatives/tableof.htm">Here.</a></p>
<p><strong>Derivatives of Vector Functions</strong><br />
If <strong>r</strong> is vector-valued function <strong>r(t)</strong> = f(t)<strong>i</strong> + g(t)<strong>j</strong> + h(t)<strong>k</strong>, then</p>
<p><strong>r'(t)</strong> = f'(t)<strong>i</strong> + g'(t)<strong>j</strong> + h'(t)<strong>k</strong></p>
<p><strong>Tangent and Normal Vectors</strong><br />
Tangent Vector <strong>T(t)</strong>=<strong>r'(t)</strong>/<strong>abs(<strong>r'(t)</strong>)</strong>, <strong>r'(t)</strong> != <strong>0</strong>.</p>
<p>Normal Vector <strong>N(t)</strong>=<strong>T'(t)</strong>/<strong>abs(<strong>T'(t)</strong>)</strong>.</p>
<p><strong>Directional Derivative</strong><br />
If <em>f</em> is a differentiable function of x and y, then the directional derivatives of <em>f</em> in the direction of a unit vector <em>u = cos θi + sin θj</em> is</p>
<p>D<sub>u</sub>f(x,y) = f<sub>x</sub>(x,y)cosθ + f<sub>y</sub>(x,y)sinθ</p>
<p>For <em>u = ai + bj + ck</em>,</p>
<p>D<sub>u</sub>f(x,y,z) = af<sub>x</sub>(x,y,z) + bf<sub>y</sub>(x,y,z) + cf<sub>z</sub>(x,y,z);</p>
<p><strong>Gradient</strong><br /><em>∇f(x,y,z) = f<sub>x</sub>(x,y,z)i + f<sub>y</sub>(x,y,z)j + f<sub>z</sub>(x,y,z)k</em></p>
<p>The direction of maximum increase is given by ∇f(x,y,z).</p>
<p><strong>Related Rates</strong></p>
<ol><li>Create a model (function) of the behavior</li>
<li>Differentiate all variables with respect to rate (usually time)</li>
<li>Plug in known values and rates</li>
</ol><p><strong>Extrema</strong><br />
To find the extrema of function f on closed interval [a,b]:</p>
<ol><li>Take the partial derivative of each variable (e.g. f<sub>x</sub>, f<sub>y</sub> f<sub>z</sub>) in f(x,y,z). Or find the gradient of the function f(x,y,z).</li>
<li>Evaluate grad f=0. A critical number requires all partial derivatives to be zero at that coordinate.</li>
<li>Evaluate f at each end point (or infinity).</li>
<li>Evaluate f at each discontinuity.</li>
<li>The minimum or maximum of the set will be the extrema. IOW extrema only occur at critical points.</li>
</ol><p><strong>Optimization: Standard</strong></p>
<ol><li>Optimization Equation: the equation to be optimized (maximum or minimum).</li>
<li>Boundary Equation(s): boundary condition equations.</li>
<li>Make optimization equation a function of a <em>single</em> variable by substituting boundary equations into the optimization equation.</li>
<li>Find extrema. Find derivative and solve for f(x)=0.</li>
<li>Eliminate results that do not make sense. Check for minima and maxima.</li>
</ol><p><strong>Optimization: Lagrange</strong></p>
<ol><li>Optimization Equation: the equation to be optimized (maximum or minimum).</li>
<li>Boundary Equation(s): boundary condition equations.</li>
<li>Take the partial derivative of the Optimization Equation for every variable in the equation (f<sub>x</sub>, f<sub>y</sub>, f<sub>z</sub>, etc).</li>
<li>Take the partial derivative of the Boundary Equation for every variable in the equation (g<sub>x</sub>, g<sub>y</sub>, g<sub>z</sub>, etc).</li>
<li>Create the Lagrange multipliers by the following equations: <em>f<sub>i</sub>=λg<sub>i</sub>. These are the Lagrange multipliers.</em></li>
<li>Solve the system of the Lagrange multipliers PLUS the Boundary Equation. This is a system of n+1 equations and n+1 variables, where n is the number of variables in the Optimization Equation.</li>
<li>Plug the coordinate from the solution above into the Optimization Equation to find the optimum value.</li>
</ol><p><strong>Next Up</strong><br />
Integration</p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/13">Hardware & Design</a></div></div></div>Sat, 01 Jul 2017 17:25:10 +0000will126 at http://teramari.ushttp://teramari.us/node/126#commentsCalculus: Limits of Functions (Part 2 of x)
http://teramari.us/node/125
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Preface</strong><br />
This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.</p>
<p><strong>Introduction</strong><br />
This article will introduce limits.</p>
<p>There are many calculus references, the one I like to use is <a href="https://www.amazon.com/Calculus-Ron-Larson/dp/1285057090/">Calculus by Larson, Edwards and Hostetler</a>.</p>
<p>This also assumes you are familiar with Python or can stumble your way through it.</p>
<p><strong>Concepts</strong><br /><em>Limit: </em>the value a function approaches as it gets close to a given input coordinate.</p>
<p><em>Asymptote: </em>the value f(x) approaches as x approaches infinity if f(x) approaches a constant value.</p>
<p><strong>Formal Definition of a Limit</strong><br />
Let f be a function and let c and L be real numbers. The limit of <em>f(x)</em> as x approaches c is L iff:<br />
lim<sub>x->c-</sub>=L and lim<sub>x->c+</sub>=L (i.e. the same from the left and from the right).</p>
<p>For functions of more than one variable the definition is the same, but extended to all variables simultaneously (e.g. an area shrinks to one point, a volume shrink to one point, etc).</p>
<p><strong>Common Symptoms of Non-Existent Limits</strong></p>
<ol><li>f(x) approaches a different number from the left and right</li>
<li>f(x) increases/decreases without bound</li>
<li>f(x) oscillates</li>
</ol><p><strong>Definition of Continuity</strong><br />
A function f is continuous at c if the following three conditions are met:</p>
<ol><li>f(c) is defined</li>
<li>lim<sub>x->c</sub> f(x) exists</li>
<li>lim<sub>x->c</sub> = f(c)</li>
</ol><p>Functions can also be open on an interval or everywhere continuous (infinite).</p>
<p><strong>Properties of Limits</strong></p>
<table><tr><td>General Form</td>
<td><em>lim<sub>s->c</sub> f(x)=L; lim<sub>s->c</sub> g(x)=K</em></td>
</tr><tr><td>Constant</td>
<td><em>lim<sub>x->c</sub> b=b</em></td>
</tr><tr><td>Variable</td>
<td><em>lim<sub>x->c</sub> x=c</em></td>
</tr><tr><td>Scalar Power</td>
<td><em>lim<sub>x->c</sub> x<sup>n</sup>=c<sup>n</sup></em></td>
</tr><tr><td>Scalar Multiple</td>
<td><em>lim<sub>x->c</sub> b*f(x)=b*L</em></td>
</tr><tr><td>Sum/Difference</td>
<td><em>lim<sub>x->c</sub> [f(x) +/- g(x)]=L +/- K</em></td>
</tr><tr><td>Product</td>
<td><em>lim<sub>x->c</sub> f(x)*g(x)=L*K</em></td>
</tr><tr><td>Quotient</td>
<td><em>lim<sub>x->c</sub> f(x)/g(x)=L/K; providing K != 0</em></td>
</tr><tr><td>Power</td>
<td><em>lim<sub>x->c</sub> f(x)<sup>n</sup>=L<sup>n</sup></em></td>
</tr><tr><td>Radical</td>
<td><em>lim<sub>x->c</sub> rad<sub>n</sub>(f(x))=rad<sub>n</sub>(L)</em></td>
</tr></table><p>Where b, c, n, L and K are constants, x is a variable and <em>providing the limit exists</em>.</p>
<p><strong>Limits of Vector Functions</strong><br />
If <strong>r</strong> is vector-valued function <strong>r(t)</strong> = f(t)<strong>i</strong> + g(t)<strong>j</strong> + h(t)<strong>k</strong>, then</p>
<p>lim<sub>t->a</sub><strong>r(t)</strong> = [lim<sub>t->a</sub> f(t)<strong>i</strong>] + [lim<sub>t->a</sub> g(t)<strong>j</strong>] + [lim<sub>t->a</sub> h(t)<strong>k</strong>]</p>
<p><strong>Next Up</strong><br />
Derivatives</p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/13">Hardware & Design</a></div></div></div>Sat, 03 Jun 2017 22:35:50 +0000will125 at http://teramari.ushttp://teramari.us/node/125#commentsCalculus: Functions (Part 1 of x)
http://teramari.us/node/124
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Preface</strong><br />
This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.</p>
<p><strong>Introduction</strong><br />
This article will introduce functions.</p>
<p>There are many calculus references, the one I like to use is <a href="https://www.amazon.com/Calculus-Ron-Larson/dp/1285057090/">Calculus by Larson, Edwards and Hostetler</a>.</p>
<p>This also assumes you are familiar with Python or can stumble your way through it.</p>
<p><strong>Concepts</strong><br /><em>Number: </em> A mathematical object or symbol used to count, measure and optionally label (interpretation).</p>
<p><em>Vector: </em> A number that indicates direction and magnitude. A vector has an origin and destination and therefore direction and magnitude.</p>
<p><em>Set: </em> a combination of numbers or variables (see below). In the form of <em>(1,2,3,...,n) or (a,b,c,...,n)</em> and usually (x,y) and (x,y,z).</p>
<p><em>Number System: </em> A set of numbers, often subjected to specific rules.</p>
<p><em>Variable: </em>a symbol used to hold a value or coordinate.</p>
<p><em>Coordinate: </em>a particular value taken on by a set used to indicate the position of a point, line, or plane. </p>
<p><em>Coordinate System: </em>usually a method of representing coordinates geometrically. However a coordinate can be represented in any coordinate system and not necessarily one that has a physical representation, in other words a coordinate system is a set of variables.</p>
<p><em>Cartesian Coordinate System: <em>a rectilinear system in which the horizontal line on a graph is the x axis and the vertical is the y axis of a plane. This can be extended to three dimensions by including the z axis to represent space extending out from the x and y axis, perpendicular to both.</em></em></p>
<p><em>Polar/Spherical Coordinate System: <em>a coordinate system that is represented by a) distance from origin, b) angle from 0 degrees (counterclockwise) and in three dimensions c) the angle from 0 degrees (and also perpendicular to both a and b).</em></em></p>
<p><em>Function: </em> a real-valued function <em>f</em> maps <em>X</em> onto <em>Y</em> such that for every coordinate in <em>X</em> there is one and one only output on <em>Y</em>; where <em>X</em> and <em>Y</em> are sets of real numbers such that <em>X=(a1,b1,c1,...,z1)</em> and <em>Y=(a2,b2,c2,...,z2)</em>.</p>
<p><em>Vector Valued Function: </em> describes the magnitude and direction of a vector originating at the origin (0,0) and terminating at the evaluation of <strong>r(t)</strong>.</p>
<p><em>Vector Field: </em> describes a plane or space on which a vector maps to every point in the space. In other words each point in the plane or space has a vector of specific magnitude and direction.</p>
<p><em>Conservative Vector Field: </em> describes a vector field <strong>F</strong> such that <strong>F</strong>=∇ <em>f</em>. In other words <strong>F</strong> is a gradient. <em>f</em>is referred to as the potential function.</p>
<p><em>Intercept: </em>a coordinate at which one coordinate value in the set is zero (e.g. f(a,b,0,d) or g(0,b,c)).</p>
<p><em>Intersection: </em>a coordinate at which two sets of data (or functions) intercept.</p>
<p><em>Graph: </em>a visual representation of coordinate sets, most often in 2 to 4 variables.</p>
<p><em>Slope: </em>rate of change of a function. For example the slope of a straight line: <em>m=(y<sub>2</sub>-y<sub>1</sub>)/(x<sub>2</sub>-x<sub>1</sub>), x<sub>1</sub> != x<sub>2</sub></em></p>
<p><em>Inverse: </em>A function g is the inverse of the function f if f(g(x)) = x for each x in the domain of g AND g(f(x)) = x for each x in the domain of f. All inverse functions are reflective about the line y=x.</p>
<p><strong>Vector and Parametric Functions</strong></p>
<table><tr><td>General Form: Vector Valued Function</td>
<td><strong>r(t)</strong>=f(t)<strong>i</strong>+g(t)<strong>j</strong>+h(t)<strong>k</strong></td>
</tr><tr><td>General Form: Vector Field</td>
<td><strong>F(x,y)</strong>=x<strong>i</strong>+y <strong>j</strong>+z<strong>k</strong><br />
x=M(x,y,z), y=N(x,y,z), z=P(x,y,z)</td>
</tr><tr><td>General Form: Parametric Form</td>
<td>x=M(t), y=N(t), z=P(t)</td>
</tr><tr><td>Addition</td>
<td><strong>u(t) + v(t)</strong>=<u<sub>i</sub> + v<sub>i</sub>, u<sub>j</sub> + v<sub>j</sub>></td>
</tr><tr><td>Scalar Product</td>
<td>c<strong>u(t)</strong>=<cu<sub>i</sub>,cu<sub>j</sub>></td>
</tr><tr><td>Dot Product</td>
<td></td>
</tr><tr><td>Cross Product</td>
<td></td>
</tr><tr><td>Commutative</td>
<td><strong>u + v</strong>=<strong>v + u</strong></td>
</tr><tr><td>Associative</td>
<td><strong>(u+v)+w</strong>=<strong>u+(v+w)</strong></td>
</tr><tr><td>Additive Identity</td>
<td><strong>u+0</strong>=<strong>u</strong></td>
</tr><tr><td>Additive Inverse</td>
<td><strong>u+(-u)</strong>=<strong>0</strong></td>
</tr><tr><td>Distributive</td>
<td>(c+d)<strong>u</strong>=c<strong>u</strong>+d<strong>u</strong></td>
</tr><tr><td>Distributive 2</td>
<td>c(<strong>u+v</strong>)=c<strong>u</strong>+c<strong>v</strong></td>
</tr><tr><td>Scalar</td>
<td>1(<strong>u</strong>)=<strong>u</strong></td>
</tr><tr><td>Scalar 2</td>
<td>0(<strong>u</strong>)=<strong>0</strong></td>
</tr><tr><td>Curl</td>
<td></td>
</tr><tr><td>Divergence</td>
<td></td>
</tr></table><p>Where f, g and h are functions of t; r is a vector function of t; and i, j and k are vectors.</p>
<p>All the properties of the functions below can be components of a vector function. Therefore all the properties of the above properties apply to the functions below.</p>
<p><strong>Linear Functions</strong></p>
<table><tr><td>General Form</td>
<td><em>Ax+By+C=0</em></td>
</tr><tr><td>Vertical Line</td>
<td><em>x=a</em></td>
</tr><tr><td>Horizontal Line</td>
<td><em>y=b</em></td>
</tr><tr><td>Point-slope Form</td>
<td><em>y-y<sub>1</sub>=m(x-x<sub>1</sub>)</em></td>
</tr><tr><td>Slope-intercept Form</td>
<td><em>y=mx+b</em></td>
</tr></table><p>Where A, B, C, y<sub>1</sub>, x<sub>1</sub>a and b are coordinates; m is the slope; x and y are variables.</p>
<p><strong>Log Functions</strong></p>
<table><tr><td>General Form</td>
<td><em>ln x = ∫<sup>b</sup><sub>a</sub> (1/t) dt, for x>0</em></td>
</tr><tr><td>1</td>
<td><em>ln(1) = 0</em></td>
</tr><tr><td>Multiple</td>
<td><em>ln(ab) = ln(a) + ln(b)</em></td>
</tr><tr><td>Power</td>
<td><em>ln(a<sup>n</sup>) = n ln a</em></td>
</tr><tr><td>Division</td>
<td><em>ln(a/b) = ln(a) - ln(b)</em></td>
</tr></table><p>Where a, b and n are coordinates; x is a variable.</p>
<p><strong>Exponential Functions</strong></p>
<table><tr><td>General Form</td>
<td><em>e<sup>x</sup> = f<sup>-1</sup>[ln(x)]</em></td>
</tr><tr><td>Inverse 1</td>
<td><em>ln(e<sup>x</sup>) = x</em></td>
</tr><tr><td>Inverse 2</td>
<td><em>e<sup>ln(x)</sup> = x</em></td>
</tr><tr><td>Multiple</td>
<td><em>e<sup>a</sup>e<sup>b</sup>=e<sup>a+b</sup></em></td>
</tr><tr><td>Division</td>
<td><em>e<sup>a</sup>/e<sup>b</sup>=e<sup>a-b</sup></em></td>
</tr></table><p>Where a, b and n are coordinates; x is a variable.</p>
<p><strong>Function Transformations</strong></p>
<table><tr><td>Original Graph (Reference)</td>
<td><em>y=f(x)</em></td>
</tr><tr><td>Horizontal Shift</td>
<td><em>y=f(x±c)</em></td>
</tr><tr><td>Vertical Shift</td>
<td><em>y=f(x)±c</em></td>
</tr><tr><td>Reflection (about x)</td>
<td><em>y=-f(x)</em></td>
</tr><tr><td>Reflection (about y)</td>
<td><em>y=f(-x)</em></td>
</tr><tr><td>Reflection (about origin)</td>
<td><em>y=-f(-x)</em></td>
</tr></table><p><strong>Coordinate Systems: Change of Variables</strong><br />
For any two coordinate systems in which the conversion between the two is defined, you can use a Jacobian Matrix to determine the conversion factor to define a change of variables.</p>
<p>If x=g(u,v) and y=h(u,v), then the Jacobian of x and y with respect to u and v is:</p>
<p><a href="https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant">The Jacobian is defined here (not easy to represent in HTML)</a></p>
<p>The change of variables for a double integral is then defined as:<br />
∫<sub>R</sub>∫ f(x,y) dx dy = ∫<sub>S</sub> ∫ f( g(u,v), h(u,v) ) det<sub>Jm</sub> du dv</p>
<p>The definition above is for two variables, but can be extended to more variables with a larger Jacobian/number of integrals and also reduced to a single variable/integral.</p>
<p><strong>Axis Intercepts</strong><br />
To determine where a function crosses the x or y axis evaluate the following function:</p>
<p>f(0) (for y-intercepts)<br />
f(x)=0 (for x-intercepts)</p>
<p>If the above formula fails try using Newton's Method (not illustrated here).</p>
<p><strong>Next Up</strong><br />
Continuity and Limits of functions.</p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/13">Hardware & Design</a></div></div></div>Sun, 21 May 2017 16:25:07 +0000will124 at http://teramari.ushttp://teramari.us/node/124#commentsMulti Party System
http://teramari.us/node/123
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>The American 2 party system is long due for a replacement. Something that allows voice, change and something other than the words "Democrat" and "Republican" which essentially have become fairly vague and meaningless.</p>
<p>But I love enjoy the drama and the process of electing US Presidents. I don't consider it a handicap that we have a long time to get to know the candidates.</p>
<p>But I have thought of a system that would fix our election system while keeping the best parts. As it turns out it is almost exactly like the European systems I've seen:</p>
<p><em>Part 1: Party Head Selection</em><br />
First, virtually an infinite number of parties are allowed. There would be some minimum requirements to being on a ballot and all state ballots (for President) will be the same.</p>
<p>Party heads can be selected any way you want. But ONLY party members select party heads. Republicans can have their own method, Democrats another. It doesn't have to be voting, open, or anything. It's up to the party members. If party members don't like it then don't be a member of that party. Join another one.</p>
<p>Once party heads are selected then televised debates can be conducted.</p>
<p><em>Part 2: Primary Election</em><br />
Primary election proceeds as normal. Except instead of using this time to select party heads this is the actual election. There is no longer a General election.</p>
<p>Difference here is that in the case of a <50% majority then the top 2 candidates head to run off.</p>
<p><em>(Optional) Part 3: Run Off Election</em><br />
A second election to choose between the two most popular candidates.</p>
<p>I'd like to point out that this does not address the electoral college. This is just one fix to a very large, complicated, broken system.</p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/1">Society & Politics</a></div></div></div>Sun, 14 May 2017 18:38:42 +0000will123 at http://teramari.ushttp://teramari.us/node/123#commentsFilm Reviews
http://teramari.us/node/122
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>I don't study film theory or anything of that sort but I like to watch films fairly critically. There has always been several things that stick out to me, like shot length, lighting and editing. Those are my favorite things. Except awesome displays of force, cool sci-fi stuff and amazing dialogue. A masterpiece takes hitting everything just right without major flaws. But I don't like writing off a film entirely for one major flaw, even if it is dialogue or plot, as painful as those two are when missing. Instead I try to look for what a film does right. So I jotted down my thoughts and organized them into the cool circle below:</p>
<p><img height="586" width="570" src="http://www.teramari.us/files/film/film.png" /></p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/15">Film</a></div></div></div>Sat, 13 May 2017 23:42:50 +0000will122 at http://teramari.ushttp://teramari.us/node/122#commentsRugged Watch Criteria
http://teramari.us/node/121
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Watches are jewelry and as a result there is a wide variation both in price and quality. I'd love to perform a study and scatter map price and quality. The most you could hope for is a straight line: higher price results in higher quality. But in reality I think what you'd end up with is a cloud with very little statistical correlation. Even if you were able to normalize for precious metals or technologies that lend themselves to aesthetics rather than robustness I don't think if would make much of a difference.</p>
<p>That hasn't stopped me from attempting the task for myself. There are several watch brands that spec (and in some cases overspec) their watches. Some invest heavily in new technologies, some use custom movements, some don't. Some are just a little more honest in what you are buying. On the opposite end are many watches that make no claims about anything and are modestly priced. Still others make grandiose claims about performance and charge for it. Sometimes those watch brands back it up and some don't.</p>
<p>So what I have done is cooked up a list of criteria, a specification if you will, for watch performance. And to make it easy to compare watches I have created a scoring system as well.</p>
<p>Watch design is essentially a two step process:</p>
<ol><li>Choose a watch movement with some minimum nominal performance.</li>
<li>Design structure around the movement that minimizes any environmental variation from nominal to ensure performance does not vary.</li>
</ol><p>IMHO analog watch movements are pretty good. They aren't as accurate as quartz and for that reason I prefer quartz most of the time. But I also think that the analog movement industry has stood still for too long. They are offering the same 3-5 9's quality of movements for literally decades now without much improvement. I don't wish to be too harsh. ETA has introduced the first affordable perpetual movement. And many movement designers are introducing these "double barrel" movements which increase the reserve to around 80 hours. But still Omega is the only vendor who is really addressing watch accuracy. No ultra stable watches have been produced in the analog segment. At least better than COSC. I know it is a herculean task but no public efforts made that I can find.</p>
<p>Accuracy is straightforward once you understand the concepts:</p>
<ul><li><em>Accuracy over time:</em> This is the long term accuracy of a watch. It is an average of all the fluctuations, perturbations, etc.</li>
<li><em>Max variation:</em> Accuracy will have peaks/values though. One day it might be +2, the next -10. That still makes the average -6. There should not be too many extremes. I have not listed this as a requirement for my watch guide.</li>
<li><em>Isochronism:</em> This is the term that defines how accuracy is affected as a function of the reserve. Watches typically have best accuracy at full winding (100% energy reserve) and get worse as they deplete. In other words over night or left at home for a day you can expect more drift than if it were on your hand being automatically wound and kept at 100%.
</li></ul><p>Here is the criteria for nominal watch accuracy:</p>
<table><tr><td>Requirement</td>
<td>Description</td>
<td>Moderate</td>
<td>Excellent</td>
</tr><tr><td>Nominal</td>
<td>Maximum average accuracy to be expected from a randomly manufactured movement in a specific environment (e.g. humidity, temp,<br />
no magnetism, etc)</td>
<td>+/-59s per month</td>
<td>+/-5s per month</td>
</tr><tr><td>Isochronism</td>
<td>Maximum error to be expected as a watch winds down and uses up its reserve (but of course hasn't stopped)</td>
<td>3-10s per day</td>
<td><3s per day</td>
</tr></table><p>Here are the criteria for robustness:</p>
<table><tr><td>Requirement</td>
<td>Description</td>
<td>Moderate</td>
<td>Excellent</td>
</tr><tr><td>Env: Magnetism</td>
<td>Resistance to magnetic fields. Very few things exceed ~1,500A/m</td>
<td>1,500-4,800A/m</td>
<td>>4,800A/m</td>
</tr><tr><td>Env: Humidity/Water</td>
<td>Resistance to humidity, water and low pressure</td>
<td>5-10bar</td>
<td>10bar</td>
</tr><tr><td>Env: Temperature</td>
<td>Exposure to temperature including direct sunlight</td>
<td>0-50degC</td>
<td>-10-60degC</td>
</tr><tr><td>Env: Shock/Vibe</td>
<td>Shock comes from drops and knocks (e.g. tables) and vibe comes from equipment, driving, trains, etc.</td>
<td>DIN9110/ISO1413</td>
<td>DIN8330</td>
</tr><tr><td>Env: Orientation</td>
<td>Performance over watch orientation (e.g. face up, face down, on it's side, etc)</td>
<td>4 or 5</td>
<td>ALL (i.e. 6)</td>
</tr><tr><td>Strength: Case</td>
<td>How resistant is the case to scuffs, scratches and dents?</td>
<td>100-500 Vickers</td>
<td>>500 Vickers</td>
</tr><tr><td>Strength: Coverglass</td>
<td>How resistant is the coverglass to scuffs, scratches and dents?</td>
<td>Mineral/Acrylic</td>
<td>Sapphire/Gorilla</td>
</tr></table><p>And here are some functional criteria to round out the selection:</p>
<table><tr><td>Requirement</td>
<td>Description</td>
<td>Moderate</td>
<td>Excellent</td>
</tr><tr><td>Function: Night Readability</td>
<td>How well does the lume work? How long does it last?</td>
<td>2-6hrs</td>
<td>>6hrs</td>
</tr><tr><td>Function: Energy Reserve</td>
<td>How many hours / months does the energy reserve last?</td>
<td>Auto: 3-6days<br />
Quartz: 24-42mths</td>
<td>Auto: >6days<br />
Quartz: >42mths</td>
</tr><tr><td>Function: Style</td>
<td>Subjective style requirements?</td>
<td>Subjective</td>
<td>Subjective</td>
</tr><tr><td>Function: Complications</td>
<td>Any complications I desire in a new watch</td>
<td>Subjective</td>
<td>Subjective</td>
</tr></table><p>The scoring system is weighted with the following terms:</p>
<ul><li>Excellent Performance: +1 point</li>
<li>Moderate Performance: +0.5 point</li>
<li>Weak Performance: +0.1 point</li>
<li>Unspecified: -0.1 point</li>
</ul><p>Any specification that does not meet at least Moderate is Weak. Any performance that is not specified receives a slight penalty. This is to encourage manufacturers to print more specifications and to stand by their (often very expensive) products. The top score is 13 points.</p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/7">Science & Technology</a></div></div></div>Sat, 06 May 2017 16:46:59 +0000administrator121 at http://teramari.ushttp://teramari.us/node/121#commentsIs pain a spandrel?
http://teramari.us/node/120
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Is pain an evolutionary evolved feature?</p>
<p>I can see the usefulness of a warning system that allows injury to be detected, mitigated and protected so the body can repair.</p>
<p>And for that it is very effective: any injury to the skin or flesh wounds are detectable, can be protected and will eventually heal.</p>
<p>But first remember that this system was developed well before we were conscious beings probably back to the times when fish were the most developed animals on the planet hundreds of millions of years ago. In the past if a fish, mammal or primitive man encountered pain there was literally nothing that could be done about it. And this is why it is complicated.</p>
<p>First of all if you sustained a massive injury being incapacitated by pain is simply no way to encourage survival. Yes some people go into a kind of adrenaline fueled "super human" mode but many do not. They just black out from pain or whatever. How is this useful?</p>
<p>This leads me to point two. Of what value is internal pain? Why are there pain receptors in locations that cannot be reached by pre-technology man? Before the advent of surgery there is literally nothing useful that can be done with this information.</p>
<p>Lastly, of what value is chronic pain? Perhaps as a reminder to slow down or be careful? Often times slowing down or being careful doesn't help or prevent injury so what's the use? And what about the myriad of other chronic pains that come from cancer, infection or genetic disorders? Being in constant pain only serves to make a person suffer for no reason.</p>
<p>I can't see how pain was an evolutionary advantage. Perhaps these side affects are nothing but spandrels? Perhaps chronic pain did not stop us from reproducing and we are left with a disappointing warning system.</p>
</div></div></div><div class="field field-name-taxonomy-vocabulary-1 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/14">Biology & Evolution</a></div></div></div>Sat, 29 Apr 2017 16:10:38 +0000will120 at http://teramari.ushttp://teramari.us/node/120#comments