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Mathematics - Mathematical Analysis Multivariable examples by drifter1

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Mathematics - Mathematical Analysis Multivariable examples
<html>
<p><img src="http://www.jobsxs.com/wp-content/uploads/2017/11/examples-example.png"/></p>
<p><a href="http://www.jobsxs.com/examples/examples-example/">Source</a></p>
<p><br></p>
<h2>Introduction</h2>
<p>Hello it's a me again drifter1!<br>
&nbsp;&nbsp;&nbsp;Today we will get into the promised <strong>examples/exercises</strong> for the whole Multivariable-part of my <strong>Mathematical Analysis series</strong> and so for every post starting from <a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-vectors-lines-and-planes">Vectors, Lines, Planes</a> up to <a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-surface-and-contour-integrals">Surface and Contour Integrals</a>.</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;I will try to <strong>solve them in detail</strong> so that you understand every bit of my procedure of thinking and why I apply each calculation, theorem etc.</p>
<p>Also, I will try <strong>"splitting" them based on the subtopic they are about</strong>...</p>
<p>So, without further do, let's get straight into it!</p>
<p><br></p>
<h2>Vectors, Lines and Planes</h2>
<p>Examples based on the posts:</p>
<ul>
  <li><a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-vectors-lines-and-planes">Vectors, Lines and Planes</a></li>
  <li><a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-advanced-plane-types">Advanced Plane Types</a></li>
</ul>
<p><br></p>
<p><strong>1.</strong>&nbsp;</p>
<p>Suppose we have the following Planes/Surfaces.</p>
<p>E1: 5x + y - &nbsp;z = 10</p>
<p>E2: x - 2y + 3z = -1</p>
<p>What is the angle φ(E1, E2) between them?</p>
<p><br></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;The angle is of course equal to the same angle of the normal vectors N1 and N2 on top of surface E1 and E2 correspondingly.</p>
<p>That way we will find the angle φ(N1, N2) = φ(E1, E2).</p>
<p><br></p>
<p>Using the coefficients of x, y and z for each plane we can find the corresponding N'vectors and so:</p>
<p>N1 = (5, 1, -1)</p>
<p>N2 = (1, -2, 3)</p>
<p><br></p>
<p>The crossproduct of them gives us the following determinant:</p>
<p><img src="http://quicklatex.com/cache3/d0/ql_1fd60b4bb47139215ac5df1d35e07ad0_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p>which of course is the vector: (1, -16, 11)</p>
<p>And so:</p>
<p><img src="http://quicklatex.com/cache3/ec/ql_0c357b65e5cdb299124501f956857bec_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p>Because the crossproduct is |ab| = |a||b|sinφ we find the angle like that:</p>
<p><img src="http://quicklatex.com/cache3/ec/ql_8dd0cf7f445ef18fa03c6f3af4c0ceec_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p><br></p>
<p><strong>2.</strong>&nbsp;</p>
<p>Suppose a surface E that:</p>
<ul>
  <li>passes through P(4, 2, 1)</li>
  <li>is vertical across to N(6, -2, 3)</li>
</ul>
<p>Calculate the distance of the center O(0, 0, 0) to the surface and so:</p>
<p>d(O, E) = ?</p>
<p><br></p>
<p>N is of course a normal vector of the plane and because the plane also passes from P we have:</p>
<p>6(x - 4) - 2(y - 2) + 3(z - 1) = 0 =&gt;</p>
<p>E: 6x - 2y + 3z = 23</p>
<p><br></p>
<p>Let's suppose another point P'(x0, y0, z0) on top of the plane so that OP' is parallel to N.</p>
<p>This means that:</p>
<p>OP' x N = 0 =&gt;</p>
<p><img src="http://quicklatex.com/cache3/29/ql_8cadf26b3fb5dd453dafb47229efd529_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p>And so we end up with a system of 3 linear equations:</p>
<p>3y0 - 2z0 = 0</p>
<p>3x0 - 6z0 = 0</p>
<p>-2x0 - 6y0 = 0</p>
<p><br></p>
<p>Which means that:</p>
<p>z0 = 3y0/2 = 3x0/6</p>
<p>y0 = -2x0/6</p>
<p>and so because P' is on top of E we have:</p>
<p><img src="http://quicklatex.com/cache3/ee/ql_ebbeec4f706effaa244ee5f9903e44ee_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p>By putting this values in the equations of before we end up with:</p>
<p><img src="http://quicklatex.com/cache3/c1/ql_58adf2da8352c95172a77d6fce46ddc1_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p>And because d(O, E) = d(O, P') we have:</p>
<p><img src="http://quicklatex.com/cache3/ec/ql_c816cb291d29dd408bd81958d5c72fec_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<h2>Vector functions and Partial Derivatives</h2>
<p>Examples based on the posts:</p>
<ul>
  <li><a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-multivariable-and-vector-functions">Multivariable and Vector Functions</a></li>
  <li><a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-partial-derivatives">Partial Derivatives</a></li>
  <li><a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-directional-derivatives">Directional Derivatives</a></li>
  <li><a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-total-differential">Total Differential</a></li>
</ul>
<p><br></p>
<p><strong>1.</strong>&nbsp;</p>
<p>Suppose the linear approximation of a curve r(t):</p>
<p>T = (dx/dt, dy/dt, dz/dt) = (1 + sint, 2cos(2t), -3sin(3t))</p>
<p>Find the line ε that:</p>
<ul>
  <li>is vertical across to r(t)</li>
  <li>and that passes through Pt(x(t), y(t), z(t)) for t = π/2</li>
</ul>
<p><br></p>
<p>For t = π/2:</p>
<p>T(π/2) = (1+1, -2, 3) = (2, -2, 3)</p>
<p><br></p>
<p>Because ε is vertical across it's parallel to the normal vector N.</p>
<p>We know that N = (∂r/∂x, ∂r/∂y, ∂r/∂z) and so:</p>
<p>N = (t - cost, 3tsin(2t), 1+cos(3t)) and for t = π/2 =&gt;</p>
<p>N = (π/2, 3, 1)</p>
<p><br></p>
<p>And so from the parametric representation of a line we have:</p>
<p>x = x0 + at</p>
<p>y = y0 + bt</p>
<p>z = z0 + ct</p>
<p>By setting (x0, y0, z0) = (π/2, 3, 1) we get:</p>
<p>x = π/2 + π/2t</p>
<p>y = 3+3t</p>
<p>z = 1 + t</p>
<p>or</p>
<p><img src="http://quicklatex.com/cache3/3b/ql_8eeeda4b7d5812ea2c527069c51c4f3b_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p>By using the normal vector N we can also find plane E that contains the Point Pt.</p>
<p>E: a(x - x0) + b(y - y0) + c(z - z0) = 0 =&gt; ... =&gt;</p>
<p><img src="http://quicklatex.com/cache3/37/ql_7a7630154fffb656684d0704373b7c37_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p><strong>2.</strong>&nbsp;</p>
<p>Suppose the function:</p>
<p><img src="http://quicklatex.com/cache3/e0/ql_eee19c2bd0e860d8018f3c90669b33e0_l3.png"/></p>
<p>Calculate the range of Duf(P) = ∇T(P) * u at P = (0, 0)</p>
<p><br></p>
<p>We know that Duf(P) is between two values:</p>
<ol>
  <li>The highest possible increase |∇f(P)| and</li>
  <li>The highest possible decrease -|∇f(P)|</li>
</ol>
<p>cause Duf(P) = ∇f(P) * u = |∇f(P) ||u|*cosφ</p>
<p>and so the highest increase is when φ =0 and the highest decrease when φ = π</p>
<p><br></p>
<p>&nbsp;∇T(x, y) = (∂T/∂x, ∂T/∂y) = <img src="http://quicklatex.com/cache3/d4/ql_5f4352cc61c0a7a8393bc1aff8f7c8d4_l3.png"/></p>
<p>For the point (x, y) = (0, 0) this becomes: (1, 1)</p>
<p><br></p>
<p>And so the highest possible increase is:</p>
<p><img src="http://quicklatex.com/cache3/3e/ql_f8e642d813aa3d8813fb23437876aa3e_l3.png"/></p>
<p><br></p>
<p>The highest possible decrease is of course the negative of that...</p>
<p>And so the range of Duf(P) is:</p>
<p><img src="http://quicklatex.com/cache3/ef/ql_816e3376601e99198715d7f4eb3229ef_l3.png"/>&nbsp;</p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p><br></p>
<p><strong>3.</strong>&nbsp;</p>
<p>Suppose we have the total differential:</p>
<p><img src="http://quicklatex.com/cache3/72/ql_993afbf68fd4322eb4a5b450533daf72_l3.png"/></p>
<p>Find a function f (if it exists) that has this df.</p>
<p><br></p>
<p>From theory we know that:</p>
<p>df = Mdx + Ndy represents the total differential of a function f only if:</p>
<p>My = Nx &lt;=&gt; ∂M/∂y = ∂N/∂x</p>
<p>that can be easily proven...</p>
<p><br></p>
<p>Let's check if that is true...</p>
<p><img src="http://quicklatex.com/cache3/b7/ql_fce40f55d80663a6ef6d7ffe7c92b9b7_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p>We can clearly see that My = Nx and so there exists an f with this df.</p>
<p><br></p>
<p>Let's now find this function...</p>
<p>M = x + e^(x,/y) = ∂f/∂x =&gt;</p>
<p><img src="http://quicklatex.com/cache3/4c/ql_fb5d1b1278db1ca687bd8e09de6c3c4c_l3.png"/></p>
<p><br></p>
<p>N = ∂f/∂y =&gt;</p>
<p><img src="http://quicklatex.com/cache3/5e/ql_5946206d4a760485fda4beb81a85b85e_l3.png"/></p>
<p>and so:</p>
<p><img src="http://quicklatex.com/cache3/e4/ql_5b2fc57b102b8bae7e1f3eb808a93ae4_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<h2>Surface and Contour Integrals</h2>
<p>Examples based on the posts:</p>
<ul>
  <li><a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-double-and-multiple-integrals">&nbsp;Double and Multiple Integrals</a></li>
  <li><a href="https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-surface-and-contour-integrals">&nbsp;Surface and Contour Integrals</a></li>
</ul>
<p><br></p>
<p><strong>1.</strong>&nbsp;</p>
<p>Calculate the surface integral:</p>
<p><img src="http://quicklatex.com/cache3/54/ql_a71cf2497552c988dae2731f39463254_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p>The "inner" integral doesn't contain x and so is a "constant" which means that:</p>
<p><img src="http://quicklatex.com/cache3/d6/ql_e9b0f51900c2d32e74556fb961f1eed6_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p>Because we have an subroot we substitute using:</p>
<p>y = 2sint =&gt; dy = 2costdt</p>
<p>which gets us to the range:</p>
<p>y -&gt; 0 =&gt; t -&gt; 0</p>
<p>y -&gt; 2 =&gt; t -&gt; π/2</p>
<p>and so:</p>
<p><img src="http://quicklatex.com/cache3/a6/ql_f97a460a7287af471716ba7a4f530ea6_l3.png"/></p>
<p>cos^2t = 1 + cos(2t) / 2 and so:</p>
<p><img src="http://quicklatex.com/cache3/15/ql_e98ba8ba20fb7ef69ce25108d5feaa15_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p><br></p>
<p><br></p>
<p><strong>2.</strong>&nbsp;</p>
<p>Tranform the contour integral to an Surface one:</p>
<p><img src="http://quicklatex.com/cache3/d7/ql_5f52ca025a07cb8e374d018b567191d7_l3.png"/></p>
<p><a href="http://quicklatex.com/">quicklatex</a></p>
<p><br></p>
<p>Using Green's theorem we get:</p>
<p><img src="http://quicklatex.com/cache3/99/ql_006722bdf45ffaa4a8cce0eb9a0bde99_l3.png"/></p>
<p><br></p>
<p>Changing to the polar coordinate space we can also get:</p>
<p><img src="http://quicklatex.com/cache3/30/ql_5fb67c34cf2b5bc2d74b88216552dc30_l3.png"/></p>
<p><br></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Knowing the ranges of x and y (or r and θ) and so the Area R or G we can then calculate the actual value of I.</p>
<p><br></p>
<p>And this is actually it for today and I hope that you enjoyed those examples!</p>
<p>From next time in Mathematics we will get into a new "branch" of Mathematics :)</p>
<p>Bye!</p>
</html>
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vote details (79)
@mrfelix ·
$0.09
Good approach. Your work is really great; i love all of it. I saw your recap post, it was really awesome! You have a new friend and follower! I will definitely be in touch with this blog.
👍  ,
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@drifter1 · (edited)
$0.09
Thank you! You are so kind.
Very happy to have you join my journey to the Moon...Ehmm I mean I'm glad that you enjoy my posts and I will be checking on your blog too, from time to time...
Seems like you also blog about great stuff :)
👍  ,
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@mrfelix ·
$0.07
Thank you boss.
> Seems like you also blog about great stuff :)

...woudn't know exactly what kinda contents appeal to you, but i definitely do what i gotta do, i just hope you like 'em.
👍  ,
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@prettyrose ·
$0.10
Very explicit. Just as a suggestion @drifter1...i would love you to add the immediate past publication of each particular series you write about:

something like...

PREVIOUSLY ON THIS SERIES:
[Mathematical Analysis - Surface and Contour Integrals](https://steemit.com/mathematics/@drifter1/mathematics-mathematical-analysis-surface-and-contour-integrals)

Just for ease of navigation.
👍  ,
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vote details (2)
@drifter1 ·
$0.06
Yes, I'm trying to add the previous posts of those series, but sometimes I forget :P
Maybe I should do it from now on so that it becomes an habit...
👍  
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