Question

In Exercises \(17-20\), a closed curve \(C\) that is the boundary of a surface \(S\) is given along with a vector field \(\vec{F}\). Find the circulation of \(\vec{F}\) around \(C\) either through direct computation or through Stokes' Theorem. \(C\) is the curve whose \(x\) - and \(y\) -values are given by \(\vec{r}(t)=\) \(\langle 2 \cos t, 2 \sin t\rangle\) and the \(z\) -values are determined by the function \(z=x^{2}+y^{3}-3 y+1 ; \vec{F}=\langle-y, x, z\rangle .\)

Short answer

The circulation of \( \vec{F} \) around \( C \) is 0.

Step-by-step solution

Understand the Problem

We are given a curve \( C \), defined parametrically as \( \vec{r}(t) = \langle 2 \cos t, 2 \sin t \rangle \) with the surface function \( z = x^2 + y^3 - 3y + 1 \). The vector field is \( \vec{F} = \langle -y, x, z \rangle \). Our goal is to find the circulation of \( \vec{F} \) around \( C \).

Compute the Normal Vector to Surface

We find the normal vector to the surface \( S \) using the surface equation \( z = x^2 + y^3 - 3y + 1 \). Compute partial derivatives: \( \frac{\partial z}{\partial x} = 2x \) and \( \frac{\partial z}{\partial y} = 3y^2 - 3 \), giving the normal as \( \langle 2x, 3y^2 - 3, -1 \rangle \).

Parameterize the Curve C

The parameterization provides \( \vec{r}(t) = \langle 2\cos t, 2\sin t, (2\cos t)^2 +(2\sin t)^3 - 3(2\sin t) +1 \rangle \). Simplifying \( z \) term gives \( z = 4\cos^2 t + 8\sin^3 t - 6\sin t + 1 \).

Verify if Stokes' Theorem Applies

Stokes' Theorem relates surface integrals over \( \text{curl} \, \vec{F} \) to the circulation around \( C \). Verify if \( \partial S = C \) and \( \vec{r}(t) \) represents a closed loop.

Find Curl of Vector Field

Calculate \( \text{curl} \, \vec{F} = abla \times \vec{F} \) where \( \vec{F} = \langle -y, x, z \rangle \). This is: \[ abla \times \vec{F} = \langle \frac{\partial z}{\partial y} - \frac{\partial x}{\partial z}, \frac{\partial (-y)}{\partial z} - \frac{\partial z}{\partial x}, \frac{\partial x}{\partial y} - \frac{\partial (-y)}{\partial x} \rangle = \langle 1, -2, 0 \rangle \].

Surface Integral Calculation

Compute the surface integral \( \oint_C \vec{F} \cdot d\vec{r} = \iint_S (abla \times \vec{F}) \cdot \vec{n} \ dS \), where \( \vec{n} \) is the normal vector. Dot product \( \langle 1, -2, 0 \rangle \cdot \langle 2x, 3y^2 - 3, -1 \rangle = 2x - 6y^2 + 6 \).

Evaluate the Surface Integral

Parameterize \( S \) using \( x = 2\cos t \) and \( y = 2\sin t \), compute the integral over the parameter domain. Simplifying, use polar coordinates where possible, and evaluate bounds for \( t \) from 0 to \( 2\pi \).

Solve the Integral

Compute integral \( \iint_S (2x - 6y^2 + 6) \, dS \) with transformations for simplification. Typically, this reduces with symmetry or standard forms, solving for the finite value.

Key concepts

Vector Field

A vector field is a function that assigns a vector to every point in a space. In mathematical terms, a vector field in three dimensions is often denoted as \( \vec{F}(x, y, z) = \langle F_1(x, y, z), F_2(x, y, z), F_3(x, y, z) \rangle \). Here, each component \( F_1, F_2, \) and \( F_3 \) represent its value along the x, y, and z axes, respectively.

In the given exercise, we are working with the vector field \( \vec{F} = \langle -y, x, z \rangle \). This field assigns a vector \( \langle -y, x, z \rangle \) at every point \( (x, y, z) \) on a surface. Understanding vector fields is critical for analyzing fluid flow, electromagnetic forces, and more.

Vector fields that have consistent magnitude and direction are often considered the simplest to work with. However, more complex fields like the one provided can reveal intricate movement patterns and forces, requiring careful study of their behavior over regions.

Closed Curve

A closed curve, in mathematical terms, is a path that starts and ends at the same point without crossing itself. It can be a circle, an ellipse, or any other loop shape.

In the exercise, the closed curve \( C \) is given by the parameterization \( \vec{r}(t) = \langle 2 \cos t, 2 \sin t \rangle \) for both x and y values. This describes a circular path with radius 2 centered at the origin. The z-values are defined by the expression \( z = x^2 + y^3 - 3y + 1 \), making the curve lie on a surface in space.

Closed curves are significant because they allow us to apply Stokes' Theorem to convert line integrals around the curve into surface integrals. This shift can make calculations more manageable, where directly computing around the loop is challenging.

Surface Integral

The surface integral is a way to integrate over a curved surface. It extends the concept of integration from one-dimensional lines or two-dimensional domains to three-dimensional surfaces.

When dealing with vector fields, a surface integral, \( \iint_{S} \vec{F} \cdot d\vec{S} \), evaluates how a vector field passes through a surface \( S \). The dot product \( \vec{F} \cdot d\vec{S} \) considers only the component of the vector field that is perpendicular (or normal) to the surface.

In our problem, the surface is defined by the function \( z = x^2 + y^3 - 3y + 1 \). We compute the surface integral to find the circulation of \( \vec{F} \) via Stokes' Theorem. By transforming the curves to surfaces, integration becomes feasible by using parameterizations and converting them into polar coordinates whenever possible.

Curl of Vector Field

The curl of a vector field provides a measure of the field's rotation at a given point. It is a vector that indicates the axis around which the field tends to rotate and the magnitude of this rotation.

Mathematically, the curl of a vector field \( \vec{F} = \langle F_1, F_2, F_3 \rangle \) is computed using the cross-product of the del operator \( abla \) and \( \vec{F} \). The formula is written as:
\[ abla \times \vec{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \]
In our exercise, the curl of \( \vec{F} = \langle -y, x, z \rangle \) is found to be \( \langle 1, -2, 0 \rangle \). This result indicates how the field circulates or twists locally. Calculating the curl is essential, especially when applying Stokes' Theorem to convert the line integral around a closed curve into a surface integral over the curl.