Problem 9 - Advanced Engineering Mathematics

Question

Use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. $$ \begin{aligned} &\mathbf{F}=y^{3} \mathbf{i}-x^{3} \mathbf{j}+z^{3} \mathbf{k} ; C \text { the trace of the cylinder } x^{2}+y^{2}=1\\\ &\text { in the plane } x+y+z=1 \end{aligned} $$

Step-by-Step Solution

Answer

\(\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \sqrt{3} \cdot \pi\).

Step 1: Understand Stokes' Theorem

Stokes' theorem relates a surface integral over a surface \(S\) to a line integral over its boundary \(C\). The theorem is given by \( \oint_{C} \mathbf{F} \cdot d \mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \), where \(d\mathbf{S}\) is the vector area element of the surface \(S\).

Step 2: Identify \(\nabla \times \mathbf{F}\)

Calculate the curl of \( \mathbf{F} = y^3 \mathbf{i} - x^3 \mathbf{j} + z^3 \mathbf{k} \). Using the determinant:\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ y^3 & -x^3 & z^3 \end{vmatrix} = (3x^2 + 3y^2) \mathbf{k}\]

Step 3: Determine the Surface \(S\)

The surface \(S\) is the portion of the plane \(x + y + z = 1\) that is bound by the cylinder \(x^2 + y^2 = 1\). Parameterize the plane using \(\vec{r}(x, y) = \langle x, y, 1-x-y \rangle\).

Step 4: Find the Normal Vector \(\mathbf{n}\)

For the plane \(x + y + z = 1\), the normal vector \(\mathbf{n}\) is \(\langle 1, 1, 1 \rangle\). Normalize it to get \(\mathbf{n} = \frac{1}{\sqrt{3}} \langle 1, 1, 1 \rangle\). Thus, \(d\mathbf{S} = \mathbf{n} \, dA = \frac{1}{\sqrt{3}} \langle 1, 1, 1 \rangle \, dA\).

Step 5: Evaluate the Surface Integral

Compute the surface integral: \[\iint_{S} (3x^2 + 3y^2) \frac{1}{\sqrt{3}} \, dA\]Since \(x^2 + y^2 = 1\) on the cylinder, the integral becomes:\[\iint_{S} \frac{3}{\sqrt{3}} \, dA = \sqrt{3} \iint_{S} \, dA\]The area \(A\) of the projection of \(S\) onto the \(xy\)-plane is the area of the circle \(x^2 + y^2 = 1\), which is \(\pi\). Thus, the integral is \(\sqrt{3} \cdot \pi\).

Step 6: Conclusion

By Stokes' theorem, the line integral \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\) is equal to the surface integral, and thus it evaluates to \(\sqrt{3} \cdot \pi\).

Key Concepts

Line Integral

A line integral, often referred to as a path integral, is a type of integral where a function is evaluated along a curve. Think of it as summing up function values along a curve, instead of an entire area or volume.
In the context of vector fields, a line integral calculates how much a vector field "flows" along a path. It is represented as \( \oint_{C} \mathbf{F} \cdot d \mathbf{r} \), where \( \mathbf{F} \) is the vector field and \( C \) is the path of integration.

• To compute a line integral, break down the path into small segments.
• Calculate the contribution to the integral for each segment.
• Sum these contributions together.

In Stokes' Theorem, line integrals appear naturally as we deal with conditions along the boundary of a surface.

Surface Integral

Surface integrals extend the concept of double integrals to functions defined over a surface in three-dimensional space. They are similar to line integrals but over surfaces instead of curves.
In context, we use the notation \( \iint_{S} \mathbf{G} \cdot d\mathbf{S} \), where \( S \) is the surface of integration, and \( d\mathbf{S} \) is an infinitesimal vector area element of the surface. This integral calculates the cumulative effect of a vector field over the surface.

• It's important to parameterize surfaces to help compute surface integrals.
• These integrals lead to concepts like flux through a surface or the total vector field passing through a given surface.

In our example, we used a surface integral to relate it to the line integral boundary through Stokes' Theorem.

Curl of a Vector Field

The curl of a vector field measures the rotational tendency of the field at a given point. Consider it as describing the "twisting" and turning effect of the vector field.
To compute the curl, represented as \( abla \times \mathbf{F} \), use a determinant involving unit vectors and partial derivatives:
\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_x & F_y & F_z \end{vmatrix} \]

• Calculate the partial derivatives and evaluate the determinant.
• The result is a new vector field describing the curl.

In the exercise example, finding the curl was vital to transitioning from the line integral to the surface integral via Stokes' Theorem.

Parameterization of Surfaces

To effectively handle complex surfaces, like those encountered in the exercise, we use parameterization. This process involves expressing the surface in terms of two parameters, often using vectors to describe position.
Suppose a surface is given by a plane equation like \( z = f(x, y) \). A common parameterization is:
\[ \mathbf{r}(x, y) = \langle x, y, f(x, y) \rangle \]

• Suitable for when the surface can be directly described by a function of two variables.
• Can handle arbitrary shapes by mapping a simpler parameter set to the actual surface.

In the given problem, parameterizing the intersection of a cylinder and plane allowed for the computation of a surface integral over the bounded surface, linked by Stokes' Theorem.