cosine rose
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</p> </p>
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<li> <li>
<a href="/a-average-math2551-spring25-exam1">Multivariable Calculus Exam 1 (Thomas 12.1 -> 14.2)</a> <a href="/mathin-a-cos-rose.html">Mathin' a Cos Rose</a>
<br>By Tyler Clarke<br> <br>Wherein Tyler wishes math weren't stupid.
Note: doesn't include limits of multivariable functions.
</li> </li>
<li> <li>
<a href="/physics-thatcircuitsquestion.html">That One Physics Question</a> <a href="/physics-thatcircuitsquestion.html">That One Physics Question</a>
<br>this took me way too long to solve<br><br>covers some pretty simple circuitry equations <br>this took me way too long to solve<br><br>covers some pretty simple circuitry equations
</li>
<li>
<a href="/a-average-math2551-spring25-exam1">Multivariable Calculus Exam 1 (Thomas 12.1 -> 14.2)</a>
<br>By Tyler Clarke<br>
Note: doesn't include limits of multivariable functions.
</li>
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<p class="paragraph-P1">Mathin’ a Cos Rose</p>
<p class="paragraph-P2">by Tyler Clarke</p>
<p class="paragraph-P2"> </p>
<p class="paragraph-P2">Got me a multivariable calculus exam coming up soon. Most of the material is pretty simple – lagrange multipliers, for instance (I’ll be writing something about those soon) – but I’ve got one particular problem regarding integrating with radial coordinates that’s hard and interesting.</p>
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<div class="paragraph-P2">I’m asked to find the area of a single loop of a <span class="text-T1">rose</span><span class="text-T2"> described in radial coordinates by </span>
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</math> </span><span class="text-T2">. </span><span class="text-T3">This is difficult to visualize, but luck hath smiled upon us, and Desmos supports radial coordinates:</span></div>
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<p class="paragraph-P3"><span class="text-T2">Not too bad. The maximum extent would appear to be 5, which makes sense, because cosine maxes out at 1 and we multiply by 5. I was a bit curious to see if the multiplier of theta controls the number of loops – as it turns out, the relation is a bit complicated, and I won’t get into that now.</span></p>
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<div class="paragraph-P4"><span class="text-T4">It’s looking to me like the best solution here is to integrate for </span>
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</math> </span><span class="text-T4">, to grab that petal in the first and fourth quadrants, but I’m worried that that isn’t a </span><span class="text-T5">general</span><span class="text-T4"> enough solution: other roses that I’ve got to graph might not be so conveniently situated. The more general way I can think of is integrating for all theta and dividing by the number of petals – which isn’t necessarily easily predictable for all theta, but is a damn sight better than just </span><span class="text-T5">hoping</span><span class="text-T4"> my hand-drawn sketches appropriately capture the useful quadrants. </span><span class="text-T6">Regardless, let’s do the first idea first, and see how it works out.</span><span class="text-T4"> In this case, we </span><span class="text-T5">know</span><span class="text-T4"> there are 3 petals, so the integral should be </span>
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</math> </span><span class="text-T4">. Why  </span>
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</math> </span><span class="text-T4"> instead of just  </span>
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</math> </span><span class="text-T4">? Because of the shape of the mesh we’re integrating; I won’t elaborate, but only because </span><a href="https://tutorial.math.lamar.edu/classes/calciii/dipolarcoords.aspx" class="text-Internet_20_link"><span class="text-T4">Paul's Online Math Notes</span></a> <span class="text-T7">already did.</span></div>
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<div class="paragraph-P5">Integrating this isn’t too hard. The inner integral is simply
<!--Next 'span' is a draw:frame. -->
<span id="Object6"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mrow>
<mrow>
<munderover>
<mo stretchy="false">∫</mo>
<mn>0</mn>
<mrow>
<mn>5</mn>
<mi>cos</mi>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>3</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</munderover>
<mi>r</mi>
</mrow>
<mi mathvariant="italic">dr</mi>
<mo stretchy="false">→</mo>
</mrow>
<mrow>
<msubsup>
<mrow>
<mrow>
<mfrac>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
<mn>2</mn>
</mfrac>
</mrow>
<mo fence="true" form="postfix" stretchy="true">|</mo>
</mrow>
<mn>0</mn>
<mrow>
<mn>5</mn>
<mi>cos</mi>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>3</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</msubsup>
<mo stretchy="false">→</mo>
<mfrac>
<mn>25</mn>
<mn>2</mn>
</mfrac>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>3</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</mrow>
</math> </span>, giving us
<!--Next 'span' is a draw:frame. -->
<span id="Object7"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mfrac>
<mn>25</mn>
<mn>2</mn>
</mfrac>
<mrow>
<munderover>
<mo stretchy="false">∫</mo>
<mfrac>
<mrow>
<mo stretchy="false">−</mo>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</munderover>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>3</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
<mi>d</mi>
<mi>θ</mi>
</mrow>
</math> </span>. <span class="text-T8">This is kind of a nasty trig integral, one which I very much don’t want to derive myself, but that’s what Wolfram is for! Wolfram kindly tells me that my disgusting integral is </span>
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<span id="Object8"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<msubsup>
<mrow>
<mrow>
<mrow>
<mfrac>
<mn>25</mn>
<mn>24</mn>
</mfrac>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>6</mn>
<mrow>
<mi>θ</mi>
<mo stretchy="false">+</mo>
<mi>sin</mi>
</mrow>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>6</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="true">|</mo>
</mrow>
<mfrac>
<mrow>
<mo stretchy="false">−</mo>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msubsup>
</math> </span><span class="text-T8">. Because </span>
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<span id="Object9"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mi>sin</mi>
<mrow>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</mrow>
</math> </span><span class="text-T8">, this works out nicely to… </span>
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<span id="Object10"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mfrac>
<mn>25</mn>
<mn>24</mn>
</mfrac>
<mn>6</mn>
<mrow>
<mi>π</mi>
<mo stretchy="false">=</mo>
<mfrac>
<mn>25</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mi>π</mi>
</mrow>
</math> </span><span class="text-T8">. </span><span class="text-T9">This is wrong.</span></div>
<p class="paragraph-P6"><span class="text-T10">I’ve kept the wrong work up because it’s </span><span class="text-T11">subtly</span><span class="text-T12"> wrong, and (I think) does a pretty good job of illustrating one of the most annoying problems with polar coordinates. Do you see it?</span></p>
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<div class="paragraph-P7"><span class="text-T13">The problem lies in how cosine behaves with multipliers. </span>
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<span id="Object11"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mi>cos</mi>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mi>θ</mi>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</math> </span><span class="text-T13"> makes a full trip from 1 to 0 in </span>
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<span id="Object12"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mn>0</mn>
<mo stretchy="false">≤</mo>
<mi>θ</mi>
<mo stretchy="false">≤</mo>
<mi>π</mi>
</mrow>
</math> </span><span class="text-T13">; </span>
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<span id="Object13"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mi>cos</mi>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>3</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</math> </span><span class="text-T15"> makes </span><span class="text-T16">three</span><span class="text-T15"> such trips in the same span. And our rose has three peaks.</span></div>
<!--Next 'div' was a 'text:p'.-->
<div class="paragraph-P6"><span class="text-T17">By some horrible mathemagic, integrating what should be a half-turn integrates </span><span class="text-T18">the whole pattern</span><span class="text-T17">. As it turns out, this is true for every cosine rose: regardless of the multiplier, a full span of pi (not 2pi!) covers the entire shape. Because we only want one petal, we have to cut our</span><span class="text-T18"> bounds</span><span class="text-T19"> </span><span class="text-T17">in third: </span>
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<span id="Object14"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mrow>
<mfrac>
<mn>25</mn>
<mn>2</mn>
</mfrac>
<mrow>
<munderover>
<mo stretchy="false">∫</mo>
<mfrac>
<mrow>
<mo stretchy="false">−</mo>
<mi>π</mi>
</mrow>
<mn>6</mn>
</mfrac>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
</munderover>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>3</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
<mi>d</mi>
<mi>θ</mi>
<mo stretchy="false">→</mo>
</mrow>
<mrow>
<msubsup>
<mrow>
<mrow>
<mrow>
<mfrac>
<mn>25</mn>
<mn>24</mn>
</mfrac>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>6</mn>
<mrow>
<mi>θ</mi>
<mo stretchy="false">+</mo>
<mi>sin</mi>
</mrow>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>6</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="true">|</mo>
</mrow>
<mfrac>
<mrow>
<mo stretchy="false">−</mo>
<mi>π</mi>
</mrow>
<mn>6</mn>
</mfrac>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
</msubsup>
<mo stretchy="false">→</mo>
<mfrac>
<mn>25</mn>
<mn>12</mn>
</mfrac>
<mi>π</mi>
</mrow>
</mrow>
</math> </span><span class="text-T20">. </span><span class="text-T21">And that’s right!</span></div>
<!--Next 'div' was a 'text:p'.-->
<div class="paragraph-P8"><span class="text-T17">What if we integrated the entire shape and divided by three, as mentioned above? This gives us actually the same integral, with a different multiplier and different bounds: </span>
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<span id="Object15"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
<mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mfrac>
<mn>25</mn>
<mn>2</mn>
</mfrac>
<mrow>
<munderover>
<mo stretchy="false">∫</mo>
<mfrac>
<mrow>
<mo stretchy="false">−</mo>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</munderover>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>3</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
<mi>d</mi>
<mi>θ</mi>
<mo stretchy="false">→</mo>
</mrow>
<mrow>
<msubsup>
<mrow>
<mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mfrac>
<mn>25</mn>
<mn>24</mn>
</mfrac>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>6</mn>
<mrow>
<mi>θ</mi>
<mo stretchy="false">+</mo>
<mi>sin</mi>
</mrow>
<mrow>
<mo fence="true" form="prefix" stretchy="false">(</mo>
<mrow>
<mrow>
<mn>6</mn>
<mi>θ</mi>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="false">)</mo>
</mrow>
</mrow>
</mrow>
<mo fence="true" form="postfix" stretchy="true">|</mo>
</mrow>
<mfrac>
<mrow>
<mo stretchy="false">−</mo>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msubsup>
<mo stretchy="false">→</mo>
<mfrac>
<mn>25</mn>
<mn>12</mn>
</mfrac>
<mi>π</mi>
</mrow>
</mrow>
</math> </span><span class="text-T17">. Exactly the same! Math is weird like that.</span></div>
<p class="paragraph-P8"><span class="text-T17">To avoid boundary hell in the future, I’m just going to integrate whole shapes and divide by petals. It’s much easier to wrap my head around why that actually works.</span></p>
<p class="paragraph-P8"><span class="text-T17"/></p>
<p class="paragraph-P8"><span class="text-T17">Stay tuned for Lagrange multipliers!</span></p>
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