Problem 5 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) is \(2\pi\).

Step 1: Understand Stokes' Theorem

Stokes' Theorem relates a surface integral over a surface \(S\) to a line integral around the boundary \(C\) of \(S\). It states that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot d \mathbf{S}\), where \(abla \times \mathbf{F}\) is the curl of \(\mathbf{F}\) and \(d \mathbf{S}\) is the vector area element of the surface.

Step 2: Identify the Surface for \(C\)

The curve \(C\) is given as \(x^2 + y^2 = 1\) in the \(xy\)-plane, suggesting that the surface \(S\) is the disk \(x^2 + y^2 \leq 1\) on the \(xy\)-plane (\(z = 0\)). This disk has its normal vector pointing in the positive \(z\)-direction by convention for counterclockwise orientation when viewed from above.

Step 3: Compute \(\nabla \times \mathbf{F}\)

The curl of \(\mathbf{F} = z^2 \mathbf{i} + 2x \mathbf{j} - y^3 \mathbf{k}\) is given by \(abla \times \mathbf{F} = \left( \frac{\partial (-y^3)}{\partial y} - \frac{\partial (2x)}{\partial z} \right) \mathbf{i} + \left( \frac{\partial (z^2)}{\partial z} - \frac{\partial (-y^3)}{\partial x} \right) \mathbf{j} + \left( \frac{\partial (2x)}{\partial x} - \frac{\partial (z^2)}{\partial y} \right) \mathbf{k}\). Calculate each component.

Step 4: Evaluate the Curl

Compute each component: \(0 \mathbf{i} + \left(2 - 0\right) \mathbf{j} + \left(2 - 0\right) \mathbf{k}\) gives \(abla \times \mathbf{F} = 0\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\).

Step 5: Calculate the Surface Integral

The surface integral is \(\iint_{S} (abla \times \mathbf{F}) \cdot d \mathbf{S}\), where \(d \mathbf{S} = dS \mathbf{k}\). Here, \(abla \times \mathbf{F} \cdot \mathbf{k} = 2\), so the integral becomes \(\iint_{S} 2 \, dA\), where \(dA\) is the area element of the disk.

Step 6: Compute Area of the Disk

The area \(A\) of the disk \(x^2 + y^2 \leq 1\) is \(\pi \times 1^2 = \pi\).

Step 7: Evaluate the Integral

The surface integral \(\iint_{S} 2 \, dA = 2 \times \pi \).

Step 8: Final Answer

By Stokes' Theorem, \(\oint_{C} \mathbf{F} \cdot d \mathbf{r} = 2\pi\).

Key Concepts

Line Integral

A line integral, often used in vector calculus, is a method for integrating functions over a curve. Think of it as adding up little bits of a function along a path.
Imagine you are walking along a path, and at each small step on your path, you're picking up a piece of information (like temperature, height or a force) from a vector field. You sum these values across your whole journey to determine the line integral. It is formally defined as:

• \( \oint_{C} \mathbf{F} \cdot d \mathbf{r} \)

This notation appears when

• The curve \(C\) is the path over which you are integrating.
• The vector field \(\mathbf{F}\) represents the function you evaluate along this path.
• \(d \mathbf{r}\) symbolizes small elements of the curve, like tiny steps.

In applying Stokes' Theorem, a line integral around a closed loop \(C\) can be transformed into a surface integral. Understanding line integrals helps us measure how a vector field interacts with a curve in space.

Surface Integral

A surface integral extends the concept of multiple integrals to functions over a surface. It's helpful when you want to add up quantities over a flexible surface, shaving curves into small pieces, much like a line integral, but over a two-dimensional surface.
Imagine placing a sheet over a waving flag. You might measure how strong the wind pushes up at each point on the flag. This gives you a surface integral. In Stokes' Theorem, it appears as:

• \( \iint_{S} ( abla \times \mathbf{F}) \cdot d \mathbf{S} \)

This formula involves:

• \(S\) is the surface over which you are summing projections of force.
• \(abla \times \mathbf{F}\) is the curl of a vector field, an important operator in calculus.
• \(d \mathbf{S}\) symbolizes an infinitesimal piece of the surface with an orientation.

The idea is to measure how the vector field 'swirls' or 'rotates' about the surface, thereby evaluating the surface integral. This concept is crucial for translating circulation around a boundary curve \(C\) into a sum over the surface \(S\).

Curl of a Vector Field

The curl of a vector field is a measure of its rotation at a point. You might visualize it as how much swirling or circulating a fluid demonstrates around a point. It's a core concept in vector calculus, especially in fields that involve rotation or fluid dynamics.
Mathematically, the curl of a vector field \(\mathbf{F}\) is represented as \(abla \times \mathbf{F}\). This term will tell you if and how much the vector field swirls around a given direction. For a field in three dimensions, its standard expression is:

• \(abla \times \mathbf{F} = \left( \frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}, \frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x}, \frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y} \right)\)

Understanding the curl is essential when applying Stokes' Theorem because it bridges the line integral over a curve and the surface integral over a surface. In our example exercise, calculating the curl of \(\mathbf{F}(x, y, z) = z^2\mathbf{i} + 2x\mathbf{j} - y^3\mathbf{k}\) enables us to find how much the vector field is rotating around the boundary circle \(C\).
Using this concept, Stokes’ Theorem provides an elegant connection between line integrals and surface integrals, as we translate fluid-like rotation into cumulative surface effects.