Problem 12 - Calculus Early Transcendentals

Question

Use Stokes' Theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=(z+\sin x) \mathbf{i}+\left(x+y^{2}\right) \mathbf{j}+\left(y+e^{z}\right) \mathbf{k} ; \quad C \text { is }} \\ {\text { the intersection of the sphere } x^{2}+y^{2}+z^{2}=1 \text { and the cone }} \\ {z=\sqrt{x^{2}+y^{2}} \text { with counterclockwise orientation looking }} \\ {\text { down the positive } z \text { -axis. }}\end{array} $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{\pi}{2} \).

Step 1: Understand Stokes' Theorem

Stokes' Theorem relates a surface integral of a curl of a vector field to a line integral around its boundary. Mathematically, this theorem is represented as: \[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \] where \( S \) is the surface bounded by the line integral \( C \), and \( abla \times \mathbf{F} \) is the curl of \( \mathbf{F} \).

Step 2: Calculate the Curl of \( \mathbf{F} \)

To apply Stokes' Theorem, first find the curl of \( \mathbf{F} \): \[ abla \times \mathbf{F} = \left( \frac{\partial}{\partial y}(y+e^{z}) - \frac{\partial}{\partial z}(x+y^2) \right) \mathbf{i} + \left(\frac{\partial}{\partial z}(z+\sin x) - \frac{\partial}{\partial x}(y+e^{z}) \right) \mathbf{j} + \left( \frac{\partial}{\partial x}(x+y^2) - \frac{\partial}{\partial y}(z+\sin x) \right) \mathbf{k} \] Upon calculation, \[ abla \times \mathbf{F} = -1\mathbf{i} + \cos x \mathbf{j} + 1 \mathbf{k} \]

Step 3: Identify the Surface \( S \)

C is the boundary of the surface formed by the intersection of the sphere \( x^2 + y^2 + z^2 = 1 \) and the cone \( z = \sqrt{x^2 + y^2} \). The surface \( S \) is oriented such that looking down along the positive z-axis, the boundary \( C \) is oriented counterclockwise.

Step 4: Parametrize the Surface \( S \)

Parametrize the surface \( S \) as follows: \( x = \cos \theta \sin \phi, \ y = \sin \theta \sin \phi, \ z = \cos \phi \) where \( \theta \in [0, 2\pi] \) and \( \phi \in [0, \pi/4] \), due to the intersection with the cone.

Step 5: Compute the Surface Normal \( d\mathbf{S} \)

The normal vector to the surface in parametric form is calculated through derivatives and cross products. \[ d\mathbf{S} = \left( \frac{\partial \mathbf{r}}{\partial \phi} \times \frac{\partial \mathbf{r}}{\partial \theta} \right) d\phi d\theta \] Calculate this and ensure \( d\mathbf{S} \) is correctly oriented with respect to the stipulated conditions in step 3.

Step 6: Perform the Surface Integral

With \( abla \times \mathbf{F} \) calculated and the orientation of \( d\mathbf{S} \) established, compute the surface integral: \[ \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \] This involves substituting the appropriate expressions from earlier steps. Simplify and solve the integrals over \( \phi \) and \( \theta \).

Step 7: Conclude Using Calculated Integral

Having solved the integral in the previous step, this final value represents the line integral \( \oint_{C} \mathbf{F} \cdot d \mathbf{r} \) due to Stokes' Theorem. Make sure to verify unit consistency and proper orientation in earlier steps.

Key Concepts

Surface Integral

The concept of a **Surface Integral** extends the idea of an integral to multidimensional surfaces. In calculus, the surface integral allows for calculating the flux of a vector field through a given surface. More simply, it helps in finding how much of a vector field passes through a surface. It is an integral of a function defined over a surface in three-dimensional space. The integral sums up 'little bits' of the field over an entire surface.
To compute a surface integral, we usually define or parametrize the surface, which makes complicated surfaces simpler to work with. Once we have a parameterized form, we calculate the surface integral by taking the dot product of the vector field with the unit normal vector to the surface, and then integrating over the surface.
Stokes' theorem simplifies computing these integrals by relating them to a line integral around the boundary of the surface. This equivalence becomes valuable because line integrals are often simpler to handle than surface integrals, especially when dealing with complex vector fields or surfaces.

Curl of a Vector Field

The **Curl** of a vector field provides a way to measure the field's tendency to circulate around a point, much like how a whirlpool might swirl around a vortex. It is a vector operation in three-dimensional space, representing the rotational nature of a vector field. Mathematically, for a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is given by the determinant of a 3x3 matrix: \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \] This yields a new vector field.
The components of the resulting vector indicate circulation about the respective axes, providing essential insights into the field's behavior. If the curl of \( \mathbf{F} \) is zero everywhere, the field is called irrotational.
In the context of Stokes' Theorem, computing the curl of a vector field helps in determining the surface integral, which in turn equates to the line integral around a closed curve bounding the surface.

Parametric Surface

A **Parametric Surface** involves expressing a surface using parameters, typically represented as a pair of equations in terms of two parameters, say \( u \) and \( v \). These parameters are used to trace out the surface in space. The main advantage here is simplifying complex surfaces into more manageable mathematical forms.
For example, a surface can be described by a vector function \( \mathbf{r}(u, v) = x(u, v) \mathbf{i} + y(u, v) \mathbf{j} + z(u, v) \mathbf{k} \). Each point on the surface corresponds to a particular value of the parameters \( (u, v) \).
Parametric equations are crucial in calculus, especially for computations involving surface integrals. They allow one to break down complicated surfaces into simple, defined shapes, making multivariate integration possible.
Using parametric equations, we can apply calculus operations like taking derivatives to express tangent vectors, which are essential to finding surface normals needed to compute surface integrals.