Integrate Surface Integral w/ Jacobian Transformation - Help Needed

In summary, surface integrals are a mathematical technique for calculating the total value of a function over a given surface. This involves breaking down the surface into smaller pieces and summing up their contributions. A Jacobian transformation is a tool used to convert integrals from one coordinate system to another by multiplying the function by the determinant of the Jacobian matrix. This transformation is helpful for surface integrals as it simplifies the integral and provides insight into the surface's geometry. The process for integrating surface integrals with a Jacobian transformation involves finding the appropriate transformation, multiplying by the determinant, and solving the integral using standard techniques. These integrals have various applications in physics, engineering, and other fields, including calculating flux, surface area and volume, and
  • #1
JasonRox
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Everything was going fine until I bumped into this...

(b^2*c^2*Cos[x]^2*Sin[y]^4 + a^2*c^2*Sin[y]^4*Sin[x]^2)^(1/2)

...integrate that with respect to y, for the boundaries y=0..Pi.

A Jacobian Transformation would be a good start, but I have no idea what functions I would use to simplify that in such a way to make it easier.

Any hints would be great.
 
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  • #2
Nevermind I got it.

I did a little mistake. I miss read the exponents of the previous integral.

Thanks for stopping by though.
 

FAQ: Integrate Surface Integral w/ Jacobian Transformation - Help Needed

What is a surface integral?

A surface integral is a mathematical technique used to calculate the total value of a function over a given surface. It involves breaking down the surface into smaller pieces and then summing up the contributions of each piece to get the overall value.

What is a Jacobian transformation?

A Jacobian transformation is a mathematical tool used to convert integrals from one coordinate system to another. It involves multiplying the original function by the determinant of the Jacobian matrix, which represents the scaling and distortion of the coordinates in the new system.

How does a Jacobian transformation help with surface integrals?

A Jacobian transformation is useful for surface integrals because it allows us to change the coordinates from the original surface to a simpler, more manageable surface. This can make the integral easier to solve and can also provide insight into the geometry of the surface.

What is the process for integrating surface integrals with a Jacobian transformation?

The process for integrating surface integrals with a Jacobian transformation involves first finding the appropriate transformation that will simplify the integral. Then, the original function is multiplied by the determinant of the Jacobian matrix and the integral is set up in the new coordinate system. The integral is then solved using standard integration techniques.

What are some applications of integrating surface integrals with Jacobian transformations?

Surface integrals with Jacobian transformations have many applications in physics, engineering, and other fields of science. They are commonly used to calculate flux, which represents the flow of a vector field through a given surface. They are also used to calculate surface area and volume, as well as to find the center of mass of an object with a variable density.

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