![]() ![]() The "first octant" is chosen by the region where we let $\theta$ and $\phi$ vary (if you think carefully about it you'll see that $\pi/2$ is the right choice above). Step 3 : Simplify the integrand, which involves two vector-valued partial derivatives, a cross product, and a dot product. To compute surface integrals in a vector field, also known as three-dimensional flux. Step 2 : Apply the formula for a unit normal vector. 3d vector field example Multivariable calculus Khan Academy. Describe the surface parameterized by r(u, v) ucosv, usinv, u, < u <, 0 v < 2. Step 1: Parameterize the surface, and translate this surface integral to a double integral over the parameter space. (b) An elliptic paraboloid results from all choices of u and v in the parameter domain. In this case, since $S$ is a sphere, you can use spherical coordinates and get the parametrization Figure 16.6.3: (a) Circles arise from holding u constant the vertical parabolas arise from holding v constant. Example 16.2.2: Evaluating a Line Integral. Where the double integral on the right is calculated on the domain $D$ of the parametrization $r$. In other words, the change in arc length can be viewed as a change in the t -domain, scaled by the magnitude of vector r (t). The way you calculate the flux of $F$ across the surface $S$ is by using a parametrization $r(s,t)$ of $S$ and then ![]()
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