![]() For any value of y, x ranges from x 0 to x y so the range of x is. To find bounds for the double integral over the shadow, well let x be the inner integral. Just as a sidebar, I'd stay away from the term "flux integral" because it's confusing and can be replaced with the less confusing "surface integral" (although "flux integral" probably arose from the fact that flux is a type of surface integral. The shadow is the triangle bounded by the lines x 0, y 1 and x y. So to formally answer your question above, your first and third formulas are valid methods of expressing and computing flux (a type of surface integral), but Stokes' Theorem relates a surface integral of curl to a line integral, not flux. Step 3 : Simplify the integrand, which involves two vector-valued partial derivatives, a cross product, and a dot product. Step 2 : Apply the formula for a unit normal vector. So when the questions ask you to calculate the flux integral using Stokes' Theorem it wants you to use the theorem to evaluate the surface integral, and not the flux. Step 1: Parameterize the surface, and translate this surface integral to a double integral over the parameter space. Formally, $\displaystyle \int_ \mathbf F(x,y,z) \cdot d\mathbf r= \iint_S (\nabla \times \mathbf F) \cdot \mathbf n \ dS$. Stokes' Theorem: It relates a surface integral (but a surface integral that is not flux) of a surface, to the line integral of its boundary. That is, $\displaystyle \iint_S f(x,y,z) \ dS $ where $dS = \| \mathbf n\| \ dA$ and $\| \mathbf n\| = \| T_u \times T_v \|$ for parametrized functions.įlux Integral: Another name for surface integral. It's any integral that's integrated with respect to a surface. ![]() Surface Integral: It's an extension of the double integral over a 2D region to an integral over a 2D surface in 3D. If the field F is constant over time, you can multiply the flux at. Here's the basic outline of the thing's we've been discussing:įlux: $\displaystyle \iint_S \mathbf F \cdot d\mathbf S= \iint_S \mathbf F \cdot \mathbf n \ dS $ (This is a specific example of a surface integral where f is a vector field, i.e., the surface integral of F over S.) Flux is a finite element software for low-frequency electromagnetic and thermal simulations. You can integrate flux, which means finding how much flux has crossed over a certain time.
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