Problem 1 - Thomas Calculus

Question

In Exercises \(1-6,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \(\begin{array}{l}{\mathbf{F}=x^{2} \mathbf{i}+2 x \mathbf{j}+z^{2} \mathbf{k}} \\\ {C : \text { The ellipse } 4 x^{2}+y^{2}=4 \text { in the } x y \text { -plane, counterclockwise }} \\ {\text { when viewed from above }}\end{array}\)

Step-by-Step Solution

Answer

The circulation is \( 4\pi \).

Step 1: Understanding Stokes' Theorem

Stokes' Theorem relates a surface integral of a curl of a vector field to a line integral around its boundary. It is given by: \( \int_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \), where \( C \) is the boundary of the surface \( S \), \( \mathbf{F} \) is the vector field, and \( abla \times \mathbf{F} \) is the curl of \( \mathbf{F} \).

Step 2: Find Curl of Vector Field

Calculate the curl of \( \mathbf{F} = x^2 \mathbf{i} + 2x \mathbf{j} + z^2 \mathbf{k} \). The curl \( abla \times \mathbf{F} \) is computed as: \( abla \times \mathbf{F} = \left( \frac{\partial (z^2)}{\partial y} - \frac{\partial (2x)}{\partial z} \right) \mathbf{i} + \left( \frac{\partial (x^2)}{\partial z} - \frac{\partial (z^2)}{\partial x} \right) \mathbf{j} + \left( \frac{\partial (2x)}{\partial x} - \frac{\partial (x^2)}{\partial y} \right) \mathbf{k} \).

Step 3: Simplify Curl Expression

Substituting partial derivatives, we get: \( abla \times \mathbf{F} = (0 - 0) \mathbf{i} + (0 - 0) \mathbf{j} + (2 - 0) \mathbf{k} = 2 \mathbf{k} \). Thus, the curl of \( \mathbf{F} \) is \( 2 \mathbf{k} \).

Step 4: Set Up Surface Integral

Since the curl \( abla \times \mathbf{F} = 2\mathbf{k} \), and the surface \( S \) lies in the \( xy \)-plane (where \( d\mathbf{S} = dA \mathbf{k} \)), the surface integral becomes: \( \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = \int_{S} 2 \ dA \).

Step 5: Convert Region to Appropriate Coordinates

The given ellipse \( 4x^2 + y^2 = 4 \) can be parameterized using polar coordinates. Let \( x = \cos(\theta) \) and \( y = 2\sin(\theta) \), then: \( x^2 = \cos^2(\theta) \), \( y^2 = 4\sin^2(\theta) \). The integration bounds will be \( 0 \) to \( 2\pi \).

Step 6: Solve the Integral

The area of the ellipse, using parameterization, is given by \( \int_0^{2\pi} \int_0^1 2r \ dr\ d\theta = 2 \left( \pi \cdot 1 \right) = 2\pi \).

Step 7: Finalize Circulation Calculation

The circulation of the field around the curve \( C \), using Stokes' Theorem, is thus equal to the surface integral, which is \( 2 \times 2\pi = 4\pi \).

Key Concepts

Curl of a Vector Field

The curl of a vector field provides a way to measure the rotation or "twisting" of a vector field at a point. It is a crucial concept in vector calculus, especially when using Stokes' Theorem.
To find the curl of a vector field, we need to compute the cross product of the del operator with the vector field. For a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is given by: \[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]
The resultant vector field represents the axis of rotation and the magnitude indicates the strength of rotation. Understanding the curl helps in interpreting the dynamic behavior of fields, like fluid flow and electromagnetism.

Surface Integral

Surface integrals are used to calculate the accumulation of quantities over a surface. In this context, they help relate the circulation of a vector field around a curve to the curl of the vector field over the surface bounded by the curve.
When computing a surface integral of the curl of a vector field \( \mathbf{F} \), Stokes' Theorem is extremely useful. It can transform a complicated line integral around a curve into a more manageable surface integral:\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \]
For a surface in the \( xy \)-plane, the differential surface vector \( d\mathbf{S} \) can be expressed simply as \( dA \mathbf{k} \), where \( dA \) is the area element of the surface. This simplification is particularly helpful when the field's curl has components perpendicular to the surface.

Circulation of Vector Fields

Circulation refers to the line integral of a vector field along a closed curve. It represents the total "push" experienced around the loop, much like the flow of water around a pipe.
Stokes' Theorem allows us to equate this circulation to a surface integral of the curl over a surface bound by the curve. This is particularly useful as computing the surface integral of the curl can often be simpler than computing the line integral directly.
Through an example: consider a vector field \( \mathbf{F} \) around an ellipse described by \( 4x^2 + y^2 = 4 \) in the \( xy \)-plane. Using the curl and integrating over the enclosed surface gives straightforward answers and insights into the behavior of the field.

Ellipse in Polar Coordinates

When solving problems involving ellipses, converting to polar coordinates can greatly simplify calculations. An ellipse in standard form, like \( 4x^2 + y^2 = 4 \), can be challenging in Cartesian coordinates.
To convert, parameterize using trigonometric identities. For example:

• Let \( x = a\cos(\theta) \)
• Let \( y = b\sin(\theta) \)

Given \( a = 1 \) and \( b = 2 \), the integral bounds from \( \theta = 0 \) to \( \theta = 2\pi \) cover the entire ellipse.
This method allows easier integration when calculating areas or surface integrals over elliptical regions, leveraging the simple bounds and symmetric properties of polar coordinates.