Problem 7 - Thomas Calculus

Question

In Exercises \(7-12,\) 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{equation} \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} \end{equation}

Step-by-Step Solution

Answer

The circulation is \(4\pi\).

Step 1: Understand Stokes' Theorem

Stokes' Theorem relates a surface integral over a surface \(S\) to a line integral around the boundary \(C\). It is given by the formula \( \int_C \mathbf{F} \cdot d\mathbf{r} = \int_S (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \). We need to compute the curl of \(\mathbf{F}\) and then evaluate it over a surface bounded by \(C\).

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

The curl of a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) is calculated as \(abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}\). For \(\mathbf{F} = x^2 \mathbf{i} + 2x \mathbf{j} + z^2 \mathbf{k}\), we get \(abla \times \mathbf{F} = 0 \mathbf{i} + 0 \mathbf{j} + 2 \mathbf{k}\).

Step 3: Parameterize the Surface

The boundary \(C\) is the ellipse \(4x^2 + y^2 = 4\) on the \(xy\)-plane. We select a surface \(S\) for convenience, such as the part of the plane \(z = 0\) inside the ellipse. The normal vector on this plane is \(\mathbf{n} = \mathbf{k}\).

Step 4: Set Up the Surface Integral

The surface integral becomes \( \int_S (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \int_S 2 \cdot 1 \, dS = 2 \int_S dS \). Since \(\mathbf{n} = \mathbf{k}\), it aligns with the direction of \(abla \times \mathbf{F}\).

Step 5: Evaluate the Surface Integral

Compute \( \int_S dS \), the area of the ellipse \(4x^2 + y^2 = 4\). The ellipse is a scaled circle with area \(\pi ab\), where \(a = 1\) and \(b = 2\). Thus, the area is \(2\pi\). Multiplying by 2, we have \(2 \cdot 2\pi = 4\pi\).

Step 6: Conclusion

The circulation of the field \(\mathbf{F}\) around the curve \(C\) is \(4\pi\) in the counterclockwise direction when viewed from above.

Key Concepts

Surface Integral

A surface integral is a way to sum up the effects of a function, like a vector field, over a particular surface. Imagine you were spreading a very thin layer of paint across an irregularly shaped surface. The total amount of paint needed could be thought of as a surface integral.
In Stokes' Theorem, the surface integral ties into the curl of a vector field. It's like capturing the net whirl or rotation in a vector field over a surface, in order to understand the circulation around the boundary of that surface. In the context of the exercise, the surface integral transforms complex, circling pathways around a curve into a simpler, planar analysis of the field's rotation across a specified surface, essentially reducing a line integral into a manageable surface-wise operation.

Curl of a Vector Field

The curl of a vector field is a vector that describes the rotation or swirling strength at every point in the field. It's like finding out how much a tiny paddlewheel would spin if you placed it anywhere in the fluid flowing according to the vector field. The curl tells us how the fluid or field is twisting or rotating in space.
Mathematically, for a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\), the curl is denoted as \(abla \times \mathbf{F}\) and is given by: \(\[abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}\]\).
In the exercise, this concept was applied to find the curl of the field \(\mathbf{F} = x^2 \mathbf{i} + 2x \mathbf{j} + z^2 \mathbf{k}\), resulting in a curl of \(2 \mathbf{k}\), indicating that the field has a uniform rotational effect in the direction of the \(\mathbf{k}\) axis.

Parameterization of Surfaces

When dealing with surfaces in vector calculus, we often use parameterization to describe complex surfaces using simple expressions. Think of parameterization as a way to re-describe a surface using variables, often called parameters, that move across or flow over the entire surface. This technique is especially handy for integrating over surfaces when using the likes of Stokes' Theorem.
In this task, the surface for evaluation was chosen as the plane \(z = 0\) bounded by the ellipse \(4x^2 + y^2 = 4\). For simplicity, since the surface was part of the \(xy\)-plane, the normal vector became simply \(\mathbf{k}\). This choice of parameterization made the integral calculation smooth and direct since the normal vector aligned with the direction of the curl, ensuring clear and straightforward calculations.

Line Integral

A line integral is an integral where you take a function (often a vector field) and integrate it along a curve. Imagine sliding a measuring tape along a twisted garden hose, capturing data points as you unravel its pathway. It measures the total 'accumulation' of a phenomenon, like work done by a force, along the curve.
Particularly, Stokes' Theorem uses line integrals to relate surface integrals to the boundaries (curves) of those surfaces. Within the exercise, the line integral sought to calculate the total circulation—how much the field \(\mathbf{F}\) swirls around the curve \(C\)—by transforming this challenging task into finding the curl over a surface bounded by \(C\). Thus, it beautifully simplified calculating the path-dependent integral into a straightforward surface-area calculation.