Question

In Exercises 17 and 18 , evaluate \(\int_{S} \int \operatorname{curl} \mathbf{F} \cdot \mathbf{N} d S\) where \(S\) is the closed surface of the solid bounded by the graphs of \(x=4\) and \(z=9-y^{2},\) and the coordinate planes. $$ \mathbf{F}(x, y, z)=\left(4 x y+z^{2}\right) \mathbf{i}+\left(2 x^{2}+6 y z\right) \mathbf{j}+2 x z \mathbf{k} $$

Short answer

The calculation of the exact value of the integral is highly dependent on the specifics of how the bounding surface is parametrized. However, following these steps enables one to accurately calculate the surface integral of the curl of a specific vector field over a specific surface.

Step-by-step solution

Finding the Curl of the Vector Field

The first step is to find the curl of the vector field. The curl of a vector field \( \mathbf{F} = M\mathbf{i} + N\mathbf{j} + P\mathbf{k} \) is defined as \( \nabla \times \mathbf{F} = (\frac{\partial P}{\partial y}-\frac{\partial N}{\partial z})\mathbf{i} - (\frac{\partial M}{\partial z}-\frac{\partial P}{\partial x})\mathbf{j} + (\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y})\mathbf{k} \). Using this formula and the given vector field \( \mathbf{F}(x, y, z) = (4xy+z^{2})\mathbf{i} + (2x^{2}+6yz)\mathbf{j}+2xz\mathbf{k} \), we calculate the curl of the given vector field.

Applying Stokes' theorem

The next step is to Apply Stokes' theorem. Stokes' theorem states that \( \int_{S}\int \mathbf{curl F} \cdot \mathbf{N} d \mathbf{S} = \int_{C}\mathbf{F} \cdot d\mathbf{r} \), where C is the boundary of S. The boundary of S in this case is described by the graphs \( x = 4 \) and \( z = 9 - y^{2} \) and the coordinate planes. We convert the surface integral into a line integral along this boundary.

Parameters Substitution

To evaluate the line integral, we need to parameterize the path along the boundary of S. We substitute the parametric representation into the line integral and compute the result.

Evaluate the Integral

Finally, we evaluate the line integral either by the Fundamental Theorem of Line Integrals, if the vector field F is a gradient field, or directly compute it.