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How to Calculate Integral Flux

Calculating integral flux is a topic in vector calculus requiring understanding of integration, parameterization and vector operations. Given a vector field, F, and a surface defined by some function, the surface integral of F represents the amount of fluid flowing through that surface for a given unit of time. This is also the definition of flux. The amount of fluid flowing through a surface is dependent on the relative orientations of the surface and the flow of the fluid; hence, flux is maximized when flowing in the direction of the normal vector to the surface, and goes to zero when flowing parallel to the surface. Integral flux is applicable to any real-life fluid flow situation, from plumbing to electrical current (which can be treated as a fluid).

Instructions

- 1
  Parameterize the function defining the surface, if it is not already in this form. Define this as Phi(u,v).
- 2
  Write the equation SS F(Phi(u,v)). (DPhi/Du (u,v) X DPhi/Dv (u,v)) du dv. S stands for the integral sign, F is the vector field for the problem, Phi(u,v) is the parameterized function representing the surface, . stands for the dot product, D stands for the partial derivative sign, and X is the cross-product.
- 3
  Plug in the appropriate values into the formula. Also add appropriate limits on the definite integrals for the variable u on the inner integral and v on the outer integral. Calculate the components one at a time, first finding the partial derivatives, then taking their cross-product, then the dot product of F and the result. Continue until you have the answer, leaving pi, if present in the final answer, as pi, rather than multiplying by 3.14.

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