Answered: Use Stokes' Theorem to compute 1 z? dx…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Use Stokes' Theorem to compute
1
22 dx + (xy) dy + 2020 dz,
where C is the triangle with vertices at (1,0,0), (0, 2,0), and (0,0,2) traversed in the order.

Expert Solution

Step 1

Given

$I = \oint_{C} \frac{1}{2} x^{2} + (x y) d y + 2020 d z$

where C is the triangle with vertices at $(1, 0, 0), (0, 2, 0), (0, 0, 2)$ traversed in the order

use stoke's theorem to evaluate the integral

Step 2

solution

let

$A = (1, 0, 0) B = (0, 2, 0) C = (0, 0, 2)$

first of all find

$\vec{B C} = (x_{C} - x_{B}, y_{C} - y_{B}, z_{C} - z_{B}) \vec{C A} = (x_{A} - x_{C}, y_{A} - y_{C}, z_{A} - z_{C})$

substitute the values

$\vec{B C} = (0 - 0, 0 - 2, 2 - 0) = (0, - 2, 0) \vec{C A} = (1 - 0, 0 - 0, 0 - 2) = (1, 0, - 2)$

$\vec{B C} \times \vec{C A} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & - 2 & 0 \\ 1 & 0 & 2 \end{matrix}| = \hat{i} (4 - 0) - \hat{j} (0 - 2) + \hat{k} (0 + 2) = 4 \hat{i} + 2 \hat{j} + 2 \hat{k}$

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