Problem 12 - 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} y^{3} \mathbf{i}+\mathbf{j}+z \mathbf{k}} \\\ {C : \text { The intersection of the cylinder } x^{2}+y^{2}=4 \text { and the hemisphere }} \\ {x^{2}+y^{2}+z^{2}=16, z \geq 0, \text { counterclockwise when viewed from }} \\ {\text { above }}\end{array} \end{equation}

Step-by-Step Solution

Answer

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

Step 1: Understand Stokes' Theorem

Stokes' Theorem relates a surface integral of a curl of a vector field over a surface to a line integral of the vector field over the boundary curve of the surface. It is given by:\[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S} \]where \(C\) is the boundary of the oriented surface \(S\), \(\mathbf{F}\) is the vector field, and \(abla \times \mathbf{F}\) is the curl of \(\mathbf{F}\).

Step 2: Determine the Curl of the Vector Field

To apply Stokes' Theorem, first find the curl of \(\mathbf{F}\). Given \(\mathbf{F} = x^{2} y^{3} \mathbf{i} + \mathbf{j} + z \mathbf{k}\), calculate the curl:\[ abla \times \mathbf{F} = \left| \begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \x^2y^3 & 1 & z \end{array} \right| \]This results in:\[ abla \times \mathbf{F} = \left( 0 - 1 \right) \mathbf{i} - \left( 0 - 3x^2y^2 \right) \mathbf{j} + \left( \frac{\partial}{\partial y}(z) - \frac{\partial}{\partial z}(x^2y^3) \right) \mathbf{k} = -\mathbf{i} - 3x^2y^2 \mathbf{j}\]

Step 3: Identify the Surface and Boundary

The surface \(S\) is the cap of the hemisphere \(x^2 + y^2 + z^2 = 16\) where \(z \geq 0\), which is bounded by the curve \(C\), the intersection with the cylinder \(x^2 + y^2 = 4\). The boundary \(C\) is a circle of radius 2 at the plane \(z = \sqrt{16 - 4} = \sqrt{12}\) or \(2\sqrt{3}\).

Step 4: Set Up the Surface Integral

Now set up the surface integral \(\iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S}\). For the hemisphere, the normal vector \(\mathbf{n}\) is \((x, y, z)\) since it's a part of the sphere with radius 4. For this particular parametrization, use polar coordinates:\[ x = r \cos(\theta), \quad y = r \sin(\theta), \quad z = \sqrt{16 - r^2} \]where \(0 \leq r \leq 2\).

Step 5: Calculate the Surface Integral

The integral becomes\[ \iint_S (-\mathbf{i} - 3x^2y^2 \mathbf{j}) \cdot (x \mathbf{i} + y \mathbf{j} + z \mathbf{k}) \, r \, dr \, d\theta \]Simplifying gives:\[ -x \, dr \, d\theta \, - \, 3x^2y^2y \, dr \, d\theta \]Due to symmetry and the limits of integration, only the term in \(x\) survives because the \(y\) components integrate to zero when bounded by symmetric limits.

Step 6: Evaluate the Integral

Integrate over \(r\) from 0 to 2, and \(\theta\) from 0 to \(2\pi\):\[ \int_0^{2\pi} \int_0^2 (-r \cos(\theta)) r \, dr \, d\theta = -\int_0^{2\pi} \cos(\theta) \int_0^2 r^2 \, dr \, d\theta \]The \(r\) integral is calculated as:\[ \int_0^2 r^2 \, dr = \frac{r^3}{3} \bigg|_0^2 = \frac{8}{3} \]Thus, the double integral evaluates as:\[ -\int_0^{2\pi} \cos(\theta) \cdot \frac{8}{3} \, d\theta = 0 \]because the integral of \(\cos(\theta)\) over one period is zero.

Key Concepts

Surface Integral

A surface integral is a way to integrate over a curved surface in 3D space. It is an extension of double integrals to curved surfaces, allowing us to sum up quantities like vector fields over these surfaces. When applying Stokes' Theorem, the surface integral of the curl of a vector field over a surface is considered. The surface of interest, in this case, is a hemisphere cap where the function is defined.

• The hemisphere cap: part of the sphere defined by the equation \(x^2 + y^2 + z^2 = 16\) with \(z \geq 0\).
• The boundary: formed by the intersection with the cylinder \(x^2 + y^2 = 4\).
• Evaluating a surface integral involves parametric equations to describe the surface and involves calculating with a normal vector.

Understanding surface integrals is essential when dealing with vector fields over complex geometries.

Vector Field

A vector field is a function that assigns a vector to every point in space. In the context of this problem, we are dealing with the vector field \( \mathbf{F} = x^2y^3 \mathbf{i} + \mathbf{j} + z \mathbf{k} \). This vector field defines a movement or a force at each point.

• Each component of the vector field describes a different aspect of the vectors' behavior in space: \(x^2y^3\) for the \(x\) direction, constant \(1\) for \(y\), and \(z\) for the \(z\) direction.
• In physics, vector fields can represent different quantities like electric fields, magnetic fields, and fluid velocities.
• When analyzing such fields, the pointwise vector can show how such quantities like force or velocity change across space.

Understanding vector fields helps in analyzing the behavior of points affected by the field's influence.

Curl of a Vector Field

The curl of a vector field is a vector that describes the rotation at a point in the vector field. In Stokes' Theorem, it measures the twisting of a vector field. For the vector field \( \mathbf{F} \), the curl is calculated as \( abla \times \mathbf{F} = -\mathbf{i} - 3x^2y^2 \mathbf{j} \).

• It is found using cross products and partial derivatives, providing a rotational effect of the field.
• The curl tells us how the field 'circulates' around a given point, which is instrumental in connecting the surface integral to the line integral.
• The resulting vector indicates in which direction the circulation occurs and its magnitude.

Grasping the concept of curl is critical for understanding phenomena involving rotational forces or flows in vector fields.

Line Integral

A line integral in a vector field calculates the total effect along a curve, often representing the work done by a force. Using Stokes' Theorem, the line integral around a closed curve \( C \) is equated to the surface integral of the curl over the surface \( S \) bounded by \( C \).

• The path \( C \): in this problem, it is the intersection curve of the cylinder and hemisphere.
• Line integrals involve dot products of vector fields with differential distance vectors along the path.
• This calculation is akin to "tracing" the path and summing up the interaction of the field with segments of the path.

Understanding line integrals enables accurate calculation of work or flux in physical problems, pivotal in fields like electromagnetism and fluid dynamics.