Purchase Solution

Stoke's theorem

Not what you're looking for?

Ask Custom Question

Please show complete steps.

Attachments
Purchase this Solution
Solution Summary

Stoke's theorem is examined.

Solution Preview

Please see the explanations below. Refer to the attachment to see the equations and figure better.

Stoke's Theorem says that the following integrals are equal:

?_S???×F ?dS= ???F?dr??

In other words: The surface integral of the curl of the vector function F is equal to the line integral of the vector field over its boundary.

It looks like for your problem, the surface S is a cube. This Theorem works if one of the sides of the cube is open, so I am going to assume that the bottom edge is open.

see figure in the attached file.

First let's do the surface integral.
We need to calculate the curls of A:

? ×A= |?(i&j&k@?/?x&?/?y&?/?z@A_x&A_y&A_z )|=i((?A_z)/?y-(?A_y)/?z)+j((?A_x)/?z-(?A_z)/?x)+k((?A_y)/?x-(?A_x)/?y)

For the given function:

(?A_z)/?y=0; (?A_y)/?z=y; (?A_x)/?z=-1; (?A_z)/?x=-z; (?A_y)/?x=0; (?A_x)/?y=1

So we have:

i(0 - y) + j(-1 + z) + k(0 - 1) =

? ×A = -y i +(-1 + z) j - k or in vector notation: <-y, -1 + z, -1>

Now, we look at dS in the surface integral.

dS = n ? dA

Where n is the unit normal vector to the ...

Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Probability Quiz

Some questions on probability

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

View More Free Quizzes
Related BrainMass Solutions

  • Vector integral - stokes therom

    Problem: Evaluate the line integral by Stokes theorem (clock wise as seen by a person standing at the origin). Calculate this integral by Stoke's theorem for the following C adn F (referred to right-handed cartesian coordinates).

  • Vector Functions : Stokes' Theorem, Divergence (Cylindrical and Spherical Coordinates) and Integration using Delta Functions

    95736 Vector Functions : Stokes' Theorem, Divergence Please see the attached file for the fully formatted problems. keywords: stokes, stoke's Please see the attached file for the complete solution. Thanks for using BrainMass.

  • Stokes Theorem, Curl and Positively Oriented Hemisphere

    42947 Stokes Theorem, Curl and Positively Oriented Hemisphere 7) Use Stoke's Theorem to evaluate curl F*dS S is the hemisphere oriented in the direction of the positive x-axis. 8) Use Stokes Theorem to evaluate C is the boundary of the part

  • Vector Fields, Fundamental Theorem of Line Integrals

    Verify Stoke's Theorem by evaluating as a line integral: The boundary curve C will be where the surface intersects with plane z = 0, and so will the curve , at z =0.

  • Application of Stokes Theorem

    You may want to check out the following sites: http://mathworld.wolfram.com/SurfaceIntegral.html http://mathworld.wolfram.com/CurlTheorem.html Side view: Top view: The Stoke's theorem states that for a vector function F the following

  • polarized light

    128039 Stoke's Vector Problem: Polarized Light A light beam has a Stoke's vector of S = (1, 0.4, 0.2, 0). What is the Degree of Polarization? Would you characterize the light as linearly polarized or circularly polarized?

  • Stoke's Law air density calculation

    35181 Stoke's Law problem (3 parts) Stokes' Law describing partial sedimentation is given by: Rate = [(gd^2)(Δρ)]/18η where g = 9.81 m s^-2 is gravitational acceleration. Δρ is the density difference between particles and the

  • Vector Field, Gradients, Div, Divergence, Curl and Surface Integrals

    S is the surface of the solid bounded by the cylinder and the planes z = x +2 and z = 0. 6) Use Stoke's Theorem to evaluate curl F*dS Please see the attached file for the fully formatted problems.

  • Connections among Green's, Stokes' and Divergence theorems

    35214 Green's, Divergence and Stokes theorems Describe in 5-15 lines the links and connections among Green's theorem in all forms, Stokes' theorem and the Divergence theorem.

View More Related Solutions