# stokes theorem template

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stokes’ theorem relates a surface integral of a the curl of the vector field to a line integral of the vector field around the boundary of the surface. therefore, the surface is not oriented properly if we were to choose this normal vector. the vector field $\curl \dlvf = (-1,-1,-1)$ and the normal vector $(-r,0,0)$ are pointing in a similar direction.

we give a curve $\dlc$ and expect you to compute the surface integral over some surface $\dls$ with boundary $\dlc$. this is especially easy when that plane is parallel to a coordinate plane, as in the following example. use surface $p$, parameterized by \begin{align*} \dlsp(r,\theta) = (r \cos\theta, r\sin\theta, 1) \end{align*} for $0 \le r \le 1, 0 \le \theta \le 2\pi$. then normal vector is \begin{align*} \pdiff{\dlsp}{r} \times \pdiff{\dlsp}{\theta} = (0,0,r), \end{align*} which points in the correct direction, as mentioned above.

if one coordinate is constant, then curve is parallel to a coordinate plane. (the xz-plane for above example). for let’s take a look at a couple of examples. example 1 use stokes’ theorem to evaluate ∬ example 2. use stoke’s theorem to evaluate the line integral ∮c(y+2z)dx +(x+2 z)dy +(x+2y)dz, where c is the , stokes theorem formula, stokes theorem formula, stokes’ theorem application, stokes’ theorem and divergence theorem, stokes’ theorem and green’s theorem. how do you use stokes theorem? how do you calculate stokes theorem? why do we use stokes theorem? does stokes theorem calculate flux?

see how stokes’ theorem is used in practice. example 1: from a surface integral to line integral example. verify stokes’ theorem for the surface s described above and the vector field f=<3y,4z,-6x>. let us first c has the induced positive orientation. example 1. page 13. stokes’ theorem . the projection d of , stokes’ theorem explained, stokes theorem orientation, stokes theorem orientation, verify stokes’ theorem for rectangle, stokes theorem in hindi

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