My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

Thanks Tree2Thanks
  • 2 Post By Greens
LinkBack Thread Tools Display Modes
March 20th, 2019, 08:24 AM   #1
Joined: Apr 2017
From: India

Posts: 56
Thanks: 0

Stokes Theorem

I am unable to solve this question of Stokes Theorem concept. It seems tough to be due to the presence of intersection of plane and sphere. Please help with the correct option and show the procedure. I am thankful in advance.
Attached Images
File Type: jpg 20190320_214843.jpg (80.3 KB, 6 views)
shashank dwivedi is offline  
March 20th, 2019, 09:11 AM   #2
Greens's Avatar
Joined: Oct 2018
From: USA

Posts: 19
Thanks: 13

Math Focus: Algebraic Geometry
Stokes says that:
$\displaystyle \int_C \vec{F}\cdot d\vec{r} = \iint_{D} curl(F) \cdot ndA$

In this case, $\vec{F} = yi+zj+xk$ and $curl(F) = -i-j-k$. Now, since the intersection of the the sphere and plane is a circle on the plane, the circle shares a normal with the plane.

This normal is the unit gradient of the plane. $grad(f) = i+j+k$, divide by the magnitude to get the unit vector:

$\displaystyle n=\frac{1}{\sqrt{3}}i+\frac{1}{\sqrt{3}}j+\frac{1} {\sqrt{3}}k$

Now take the dot product of the curl and the normal to receive $-\sqrt{3}$

Lastly, integrate:

$\displaystyle \iint -\sqrt{3}dA = -\sqrt{3}\iint dA$

Since the plane slices the sphere through its center, the intersection is a circle of radius $a$, therefore:

$\displaystyle -\sqrt{3}\iint dA = -\sqrt{3} \cdot \pi a^{2}$

Now take the absolute value as the question asks to receive choice b)
Thanks from topsquark and shashank dwivedi
Greens is offline  

  My Math Forum > College Math Forum > Calculus

stokes, theorem

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Stokes' Theorem Robotboyx9 Calculus 1 October 16th, 2017 06:33 AM
Stokes theorem matteamanda Calculus 1 March 10th, 2017 03:50 AM
Stokes's Theorem and Green's Theorem djcan80 Calculus 2 August 24th, 2016 07:28 PM
Stokes theorem verification. ZardoZ Applied Math 2 June 21st, 2011 05:31 AM
proof by using greens theorem or stokes theorem george gill Calculus 5 May 14th, 2011 02:13 PM

Copyright © 2019 My Math Forum. All rights reserved.