Finding the Wrong Answer with Stokes' Theorem

In summary, the conversation discusses Stokes' theorem and the process of computing a surface integral using parametrization and the curl of a vector field. The participants also point out an error in the z-component of the curl and suggest using cyclic permutation to simplify calculations.
  • #1
greg_rack
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Homework Statement
##\vec F=<x+y^2, y+z^2,z+x^2>##, C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1). Compute ##\int_{C}^{}\vec F\cdot d \vec r##
Relevant Equations
Stoke's theorem
Computable form of a surface integral
From Stokes' theorem: ##\int_{C}^{}\vec F\cdot d\vec r=\iint_{S}^{}curl\vec F\cdot d\vec S=\iint_{D}^{}curl\vec F\cdot(\vec r_u \times \vec r_v)dA ##
To get to the latter surface integral, I started by parametrizing the triangular surface in ##uv## coordinates as:
$$\vec r=<1-u-v,u,v>, 0\leq u\leq 1, 0\leq v\leq 1-u$$
I then computed the curl of the vector field, the partial derivatives in ##u## and ##v## of the above parametrization and their cross product:
$$curl\vec F=<-2z, -2x, 2(x-y)>, \vec r_u \times \vec r_v=<1,1,1>$$
Now we can dot the curl with the cross product of the partial derivatives and get to a computable form of the surface integral
$$curl\vec F(\vec r(u,v))\cdot (\vec r_u \times \vec r_v)=-2(u+v)\rightarrow -2\int_{0}^{1}\int_{0}^{1-u}(u+v)dv du$$
which leads to a wrong answer. What am I missing?
 
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  • #2
Check your curl. The ##z##-component looks funny. :smile:
 
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  • #3
Yes I agree with ergospherical, checked the curl with wolfram and the z component is wrong.
 
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  • #4
Thank you guys, I had indeed got the k hat component of the del cross F wrong!
It's cool now
 
  • #5
Fun fact: The components of the field are just cyclic permutations ##x\to y \to z \to x## so the curl must have the same property. Therefore it is sufficient to compute one of the curl components and get the rest by cyclic permutation.
 
Last edited:
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Related to Finding the Wrong Answer with Stokes' Theorem

1. What is Stokes' Theorem?

Stokes' Theorem is a mathematical theorem that relates the surface integral of a vector field over a surface to the line integral of the same vector field around the boundary of that surface.

2. How is Stokes' Theorem used to find the wrong answer?

Stokes' Theorem can be used to find the wrong answer if the vector field being integrated is not conservative, meaning that its line integral is dependent on the path taken. This can lead to incorrect results when using the theorem.

3. What are some common mistakes when applying Stokes' Theorem?

Some common mistakes when applying Stokes' Theorem include not correctly identifying the boundary of the surface, using the wrong orientation for the line integral, and not considering the direction of the normal vector.

4. How can one avoid finding the wrong answer with Stokes' Theorem?

To avoid finding the wrong answer with Stokes' Theorem, it is important to carefully identify the surface and its boundary, use the correct orientation for the line integral, and consider the direction of the normal vector. It is also helpful to double check calculations and use multiple methods to verify the result.

5. What are some real-world applications of Stokes' Theorem?

Stokes' Theorem has many practical applications in physics and engineering, such as calculating fluid flow through a surface, determining the circulation of a vector field, and finding the work done by a force on a moving object. It is also used in electromagnetism to calculate the electric and magnetic fields around a closed loop.

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