Question

Use Stokes' Theorem to evaluate $ \int_C \textbf{F} \cdot d\textbf{r} $. In each case $ C $ is oriented counterclockwise as viewed from above. $ \textbf{F}(x, y, z) = \textbf{i} + (x + yz) \, \textbf{j} + (xy - \sqrt{z}) \, \textbf{k} $, $ C $ is the boundary of the part of the plane $ 3x + 2y + z = 1 $ in the first octant

   Use Stokes' Theorem to evaluate $ \int_C \textbf{F} \cdot d\textbf{r} $. In each case $ C $ is oriented counterclockwise as viewed from above.

$ \textbf{F}(x, y, z) = \textbf{i} + (x + yz) \, \textbf{j} + (xy - \sqrt{z}) \, \textbf{k} $,
$ C $ is the boundary of the part of the plane $ 3x + 2y + z = 1 $ in the first octant

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Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart 8th Edition

Chapter 16, Problem 8

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Solved by Expert Yuki Hotta

02/14/2021

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The curl of a vector field is given by the determinant of the matrix formed by the unit vectors $\textbf{i}$, $\textbf{j}$, $\textbf{k}$, the partial derivatives $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$, $\frac{\partial}{\partial z}$, and the Show more…

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Use Stokes' Theorem to evaluate $ \int_C \textbf{F} \cdot d\textbf{r} $. In each case $ C $ is oriented counterclockwise as viewed from above. $ \textbf{F}(x, y, z) = \textbf{i} + (x + yz) \, \textbf{j} + (xy - \sqrt{z}) \, \textbf{k} $, $ C $ is the boundary of the part of the plane $ 3x + 2y + z = 1 $ in the first octant

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00:01 All right, so let's go ahead and solve this problem.

00:03 So we're given f, which is a vector 1, comma, x plus y, z, comma, x, y, minus square root of z.

00:11 And we have a path c, which is the intersection or the boundary in the first octant and the plane 3x plus 2y plus z is equal to 1.

00:25 And i've already drawn the in 3d, the blue path is what we're looking at.

00:32 That's the boundary of c.

00:33 So we want to evaluate the integral, the integral of c, f .dr.

00:45 But we're going to use stokes theorem to evaluate this thing.

00:49 So it is going to be the double integral through the surface s, where s is going to be this triangular region, curlf.

01:08 D .s.

01:12 And as usual, instead of just doing d .s, you can do curl f times the normal vector, ds.

01:25 Okay? all right.

01:29 So, let's start with finding the curl.

01:38 This is going to be the vector, x minus y, comma negative y, comma 1.

01:47 It's quite straightforward.

01:48 So i'll let you, take a look at the details by herself.

01:52 And the normal vector, because the plane is given to you, it's quite straightforward to see.

01:59 I'm not going to normalize it, but we know that this is going to be parallel to 3 comma 2, comma, 1.

02:09 Okay? so we know that curlf .n, the dot product will simplify to 3x minus 5y plus 1...

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