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wifi said:Problem:
Check Stokes's theorem for the vector function ## v=(x-y)^3 \hat{x} + (x-y)^3 \hat{y} ## using the area and perimeter shown below.
Where do I start?
wifi said:Thanks Dick. I computed the surface integral & I got 101/6. Is this correct?
wifi said:For the RHS I did the following:
##y=2x-2 \Rightarrow dy=2dx ## (1)
## \vec{v} \cdot d\vec{\ell}=(x-3)^3 \hat{x} + (x-y)^3 \hat{y} ##(2)
Is this an acceptable answer? Does it make sense that the answer is negative?
## \int \vec{v} \cdot d\vec{\ell} = \int\limits_0^1 (x-3)^3dx + \int\limits_0^1 (x-(2x-2))^32dx = -20 ##.
wifi said:I skipped the other 2 integrals because I believe they come out to be 0.
wifi said:So for the LHS of Stokes's theorem, I get:
## \nabla \times \vec{v} = (\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y})\hat{z} = 6(x-y)^2 \hat{z}## (since ## \vec{v} ## does not have a z-component)
## (\nabla \times \vec{v}) \cdot d\vec{a} = [6(x-y)^2 \hat{z}] \cdot [dx \ dy \ \hat{z}] = 6(x-y)^2dx \ dy##
## \int (\nabla \times \vec{v}) \cdot d\vec{a} = \int \int 6(x-y)^2dx \ dy##
wifi said:So that means I must integrate over y first and then x?
wifi said:I integrated wrt to y & then x & got an answer of 7. I wasn't sure if the x limits should be from 0->1 or 1->0. I picked 1->0 because that follows the counter-clockwise path, is that correct?
Stokes's Theorem for Vector Function v=(x-y)^3 is a mathematical theorem that relates the surface integral of a vector function over a closed surface to the line integral of the same function over the boundary curve of the surface. It is a fundamental theorem in vector calculus and is often used in physics and engineering applications.
The vector function v=(x-y)^3 is often used in Stokes's Theorem because it represents a conservative vector field, meaning the line integral of the function over any closed curve is equal to zero. This simplifies the calculations and makes the theorem more applicable in real-world problems.
Stokes's Theorem can be used to calculate the area of a surface by taking the surface integral of the function v=(x-y)^3 over the surface. This is based on the concept that the flux of the vector field through the surface is equal to the line integral of the function over the boundary curve of the surface, which in this case represents the perimeter of the surface.
Yes, Stokes's Theorem can be used to calculate the perimeter of a surface by taking the line integral of the function v=(x-y)^3 over the boundary curve of the surface. This is based on the concept that the line integral of a conservative vector field over a closed curve is equal to the area enclosed by the curve.
Stokes's Theorem has numerous practical applications in various fields such as fluid dynamics, electromagnetism, and engineering. It is often used to calculate the circulation of a fluid around a closed path, the work done by a magnetic field on a moving charge, and the force exerted by a fluid on a solid object. It is also used in the study of fluid flow and heat transfer in pipes and channels.