Use Stokes' Theorem to evaluate ∮C 𝐅 ·d 𝐫 𝐅(x, y, z)=-3 y^2𝐢+4 z 𝐣+6 x 𝐤 ; C is the triang

step 1 Today we are going to solve our number 8. Here function f is x y z is equal to minus 3 y square

Use Stokes' Theorem to evaluate ∮C 𝐅 ·d 𝐫 𝐅(x, y, z)=-3...

Use Stokes' Theorem to evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ $$ \begin{array}{l}{\mathbf{F}(x, y, z)=-3 y^{2} \mathbf{i}+4 z \mathbf{j}+6 x \mathbf{k} ; C \text { is the triangle in the }} \\ {\text { plane } z=\frac{1}{2} y \text { with vertices }(2,0,0),(0,2,1), \text { and }(0,0,0)} \\ {\text { with a counterclockwise orientation looking down the pos- }} \\ {\text { itive } z \text { -axis. }}\end{array} $$

Use Stokes' Theorem to evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$
$$
\begin{array}{l}{\mathbf{F}(x, y, z)=-3 y^{2} \mathbf{i}+4 z \mathbf{j}+6 x \mathbf{k} ; C \text { is the triangle in the }} \\ {\text { plane } z=\frac{1}{2} y \text { with vertices }(2,0,0),(0,2,1), \text { and }(0,0,0)} \\ {\text { with a counterclockwise orientation looking down the pos- }} \\ {\text { itive } z \text { -axis. }}\end{array}
$$

Key Concepts

Orientation

Orientation in the context of Stokes' Theorem refers to the choice of direction on the boundary curve and the corresponding normal direction on the surface. The right-hand rule is typically used to relate the orientation of the boundary curve and the surface normal. This ensures consistency between the direction of the line integral around the curve and the surface integral of the curl, which is essential for the correct application of the theorem.

Surface Integral

A surface integral extends the concept of integration to functions defined on surfaces. It involves integrating a scalar or vector function over a surface, often by parametrizing the surface with two parameters and integrating over a region in the parameter space. For vector fields, the surface integral often involves the dot product of the field (or its curl) with a unit or oriented normal vector to the surface.

Stokes' Theorem

Stokes' Theorem is a fundamental result in vector calculus that relates the line integral of a vector field around a closed curve to the surface integral of the curl of the vector field over a surface bounded by the curve. This theorem allows us to transform a potentially difficult line integral into a usually simpler surface integral when the surface and its orientation are chosen appropriately.

Curl of a Vector Field

The curl of a vector field measures the tendency of the field to induce rotation around a point. It is defined as the cross product of the del operator with the vector field. In the application of Stokes' Theorem, the curl is integrated over the surface, so computing the curl correctly is crucial for evaluating the circulation of the vector field along the curve.

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