Question

Problem 10: Use Stokes' Theorem to evaluate $\iint_S \text{curl } \vec{F} \cdot d\vec{S}$ where $\vec{F}(x, y, z) = \arctan(x^2yz^2)\hat{i} + x^2y\hat{j} + x^2z^2\hat{k}$ and $S$ is the cone $x = \sqrt{y^2 + z^2}$ for $x \in [0, 2]$ oriented in the direction of the positive $x$-axis.

Name: Problem 10: Use Stokes' Theorem to evaluate curl F⃗· dS⃗ where F⃗(x, y, z) = arctan(x^2yz^2)î + x^2yĵ + x^2z^2k̂ and S is the cone x = √(y^2 + z^2) for x ∈ [0, 2] oriented in the direction of the positive x-axis.
Uploaded: 2024-03-23T02:57:53-08:00
Duration: 2 min 7 s
Channel: Zack A
Description: Problem 10: Use Stokes' Theorem to evaluate curl F⃗· dS⃗ where F⃗(x, y, z) = arctan(x^2yz^2)î + x^2yĵ + x^2z^2k̂ and S is the cone x = √(y^2 + z^2) for x ∈ [0, 2] oriented in the direction of the positive x-axis.

          Problem 10: Use Stokes' Theorem to evaluate $\iint_S \text{curl } \vec{F} \cdot d\vec{S}$ where $\vec{F}(x, y, z) = \arctan(x^2yz^2)\hat{i} + x^2y\hat{j} + x^2z^2\hat{k}$ and $S$ is the cone $x = \sqrt{y^2 + z^2}$ for $x \in [0, 2]$ oriented in the direction of the positive $x$-axis.

Problem 10: Use Stokes' Theorem to evaluate curl F⃗· dS⃗ where F⃗(x, y, z) = arctan(x^2yz^2)î + x^2yĵ + x^2z^2k̂ and S is the cone x = √(y^2 + z^2) for x ∈ [0, 2] oriented in the direction of the positive x-axis.

Added by Michael J.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Zack A

03/23/2024

Step 1

Using the curl formula, we have: curl(F) = (∂(x^2*z^2)/∂y - ∂(x^2*y)/∂z)i - (∂(x^2*z^2)/∂x - ∂(arctan(x^2*y*z^2))/∂z)j + (∂(x^2*y)/∂x - ∂(arctan(x^2*y*z^2))/∂y)k Calculating the partial derivatives, we get: curl(F) = (-2*x*z^2)i - (2*x*z^2 - 2*x^2*y*z/(1 + Show more…

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Problem 10: Use Stokes' Theorem to evaluate ∬_(S)curl(F)·d(S) where F(x,y,z)=arctan(x^2*y*z^2)i+x^2*yj+x^2*z^2k and S is the cone x=sqrt(y^2+z^2) for x in [0,2] oriented in the direction of the positive x-axis. Problem 10: Use Stokes' Theorem to evaluate ∬_(S)curl(F)·d(S) where F(x,y,z)=arctan(x^2*y*z^2)i+x^2*yj+x^2*z^2k and S is the cone x=sqrt(y^2+z^2) for x in [0,2] oriented in the direction of the positive x-axis.