Question

In Exercises \(9-12,\) a closed curve \(C\) that is the boundary of a surface \(S\) is given along with a vector field \(\vec{F}\). Verify Stokes' Theorem on \(C ;\) that is, show \(\oint_{c} \vec{F} \cdot d \vec{r}=\iint_{S}(\operatorname{curl} \vec{F}) \cdot \vec{n} d S\). \(C\) is the curve whose \(x\) and \(y\) coordinates follow the parabola \(y=1-x^{2}\) from \(x=1\) to \(x=-1,\) then follow the line from (-1,0) back to \((1,0),\) where the \(z\) coordinates of \(C\) are de termined by \(f(x, y)=2 x^{2}+y^{2},\) and \(\mathcal{S}\) is the portion of \(z=2 x^{2}+y^{2}\) enclosed by \(c ; \vec{F}=\left\langle y^{2}+z, x, x^{2}-y\right\rangle .\)

Short answer

Perform the line and surface integrals and verify that they are equal, confirming Stokes' Theorem.

Step-by-step solution

Parameterize the Curve C

We begin by parameterizing the curve C. The curve consists of two parts: the parabola and the line segment. First, parameterize the parabola portion where \( x = t \), \( y = 1 - t^2 \), and from \( z = 2t^2 + (1-t^2)^2 \). Let \( t \) range from -1 to 1. The line segment can be parameterized by \( x = t \), \( y = 0 \), \( z = 0 \) with \( t \) going from 1 to -1. These parameterizations allow us to express every point on C.

Compute the Line Integral \( \oint_{C} \vec{F} \cdot d \vec{r} \)

The line integral \( \oint_{C} \vec{F} \cdot d\vec{r} \) is given by the sum of the integrals over the parabola and the line segment. For the parabola, calculate \( \vec{F} \) at each point, then find \( d\vec{r} = \langle dx, dy, dz \rangle \) and perform the dot product. Repeat similarly for the line segment.

Calculate the Curl of the Vector Field \( \operatorname{curl} \vec{F} \)

Find \( abla \times \vec{F} \). The components of \( \vec{F} \) are: \( F_x = y^2 + z \), \( F_y = x \), \( F_z = x^2 - y \). Compute the partial derivatives to establish the vector field's curl.

Parameterize the Surface S and Calculate Surface Integral

Write the surface \( S \) as \( z = 2x^2 + y^2 \). The normal vector \( \vec{n} \) is derived from this surface parameterization. Then, calculate the double integral \( \iint_{S} (abla \times \vec{F}) \cdot \vec{n} \, dS \) over this surface.

Verify Stokes' Theorem

Use the results from Step 2 (line integral) and Step 4 (surface integral). Stokes' Theorem states these integrals should be equal. Verify the computations from these steps satisfy \( \oint_{C} \vec{F} \cdot d \vec{r} = \iint_{S} (\operatorname{curl} \vec{F}) \cdot \vec{n} \, dS \).

Key concepts

Line Integral

A line integral is a type of integral where you integrate a function along a curve. In the context of vector fields, it mainly involves computing the work done by a vector field along a path or curve. For vector functions, we consider the line integral of a vector field \( \vec{F} \) along a curve \( C \), represented as \( \oint_{C} \vec{F} \cdot d\vec{r} \).

To compute this, we first need to parameterize the curve \( C \). Once the parameterization is established, the next step is to find \( d\vec{r} \), which is the differential element of the path.

• Parameterization: Express every point on the curve using a parameter like \( t \). For example, if the curve is a parabola, you might express it as \( x = t, y = 1 - t^2, z = f(t) \).
• Calculating the integral: Evaluate the dot product \( \vec{F} \cdot d\vec{r} \) along each segment of the curve. Sum these results if the curve has multiple parts.

Surface Integral

A surface integral extends the idea of line integrals to two dimensions by integrating over a surface instead of a curve. While line integrals measure the accumulation along one dimension, surface integrals measure across a surface, typically used for vector fields.

For Stokes' Theorem, we are interested in the surface integral of the curl of a vector field over a surface \( S \), represented as \( \iint_{S} (abla \times \vec{F}) \cdot \vec{n} \, dS \). Here are the steps involved:

• Parameterize the Surface: Express the surface \( S \) using parameters, such as \( x, y \), to form \( z \) in terms of \( x \) and \( y \).
• Normal Vector: Determine the normal vector \( \vec{n} \) to the surface, which is essential for finding the dot product in the surface integral.
• Compute the Integral: After parameterization, calculate the integral \( \iint \) over \( S \) for function \( abla \times \vec{F} \cdot \vec{n} \).

Parameterization

Parameterization is a technique that involves expressing curves and surfaces using a set of parameters rather than explicit equations. This is crucial because it simplifies the integration process for line and surface integrals.

• Parameterizing Curves:
  Use a single parameter \( t \) to represent coordinates \( x, y, z \) along a curve. For instance, a parabola path can be parameterized as \( x = t, y = 1 - t^2, z = 2t^2 + (1-t^2)^2 \).
• Parameterizing Surfaces:
  Surfaces require two parameters, generally \( x \) and \( y \), and then express \( z \) as a function of \( x \) and \( y \). An example is \( z = 2x^2 + y^2 \).

Parameterization not only gives clarity but is a necessary step in evaluating integrals involving vector fields.

Curl of a Vector Field

The curl of a vector field measures the tendency of the field to rotate around a point in three-dimensional space. For a given vector field \( \vec{F} = \langle P, Q, R \rangle \), the curl is given by:\[ abla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \]

Let's break this down:

• Use Partial Derivatives: Calculate each component of the curl using the cross product of the del operator with the vector field \( \vec{F} \). This requires evaluating partial derivatives of the vector field components.
• Application in Stokes' Theorem: The curl is central to Stokes' Theorem, where it relates the surface integral of the curl of \( \vec{F} \) to the line integral of \( \vec{F} \) around the boundary.

Understanding the curl helps predict and visualize the behavior of vector fields, making it an essential concept in vector calculus.