Problem 18 - Calculus: Early Transcendental Functions

Question

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the boundary of the portion of \(z=x^{2}+y^{2}\) below \(z=4\) oriented downward, \(\mathbf{F}=\left\langle x^{2}, y^{4}-x, z^{2} \sin z\right\rangle\)

Step-by-Step Solution

Answer

The result of the double integral gives the result of the line integral through Stokes' theorem.

Step 1: Calculate the curl of the vector field

First of all, we need to find the curl of vector field F. The curl of a vector field \(\mathbf{F}=\langle f, g, h\rangle\) is defined as \(\nabla \times \mathbf{F} = \langle \frac{\partial h}{\partial y} - \frac{\partial g}{\partial z}, \frac{\partial f}{\partial z} - \frac{\partial h}{\partial x}, \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y}\rangle\). Substituting f, g and h with the respective components of F and calculating the partial derivatives, we get the curl of \(\mathbf{F}\) as \(\langle 2z \cos z +1, 2x, 4y^3 - 2x \rangle\).

Step 2: Parametrize the surface

We know that the surface is defined by \(z = x^2 + y^2\) and \(z\leq4\). This surface can be parametrized in polar coordinates as \(\mathbf{r}(r, \theta) = \langle r \cos(\theta), r \sin(\theta), r^2 \rangle\) where the limits for r and \(\theta\) are \(0 \leq r \leq 2\) and \(0 \leq \theta \leq 2\pi\) respectively.

Step 3: Evaluate the Surface Integral

With the parameterization and the curl of F, we can evaluate the line integral over C as a double integral over the surface S using Stokes' theorem which states: \(\int_C \mathbf{F} \cdot d \mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d \mathbf{S}\). In our case, we will have \(\iint_S (\nabla \times \mathbf{F}) \cdot d \mathbf{S} = \int_{0}^{2\pi} \int_{0}^{2} \langle 2z \cos z +1, 2x, 4y^3 - 2x \rangle \cdot \langle r \cos \theta, r \sin \theta, r\rangle rdrd\theta. \). Solving this double integral we get the answer.

Key Concepts

Curl of a Vector Field

To understand the core concepts behind Stokes' Theorem, it's crucial to delve into what the curl of a vector field is. The curl is a vector that describes the rotation of a field. Imagine you have a small paddle wheel placed in a flow of air or water. The curl at that point tells us how much the wheel would spin. Mathematically, if you have a vector field \(\mathbf{F} = \langle f, g, h \rangle\), its curl is given by:

• \(abla \times \mathbf{F} = \left\langle \frac{\partial h}{\partial y} - \frac{\partial g}{\partial z}, \frac{\partial f}{\partial z} - \frac{\partial h}{\partial x}, \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right\rangle\).

By calculating the curl, we are essentially assessing how much and in which direction a vector field is "twirling". In our problem, the vector field \(\mathbf{F}\) is defined and we've derived its curl as \(\langle 2z \cos z +1, 2x, 4y^3 - 2x \rangle\). This expression tells us how \(\mathbf{F}\) curls at any given point in space.

Parametrization of Surfaces

Parametrization is a method to express a surface as a set of equations or relations. This step is essential when calculating integrals over complex surfaces. When dealing with the surface defined by \(z = x^2 + y^2\) for a region, using polar coordinates offers a simpler approach. Switching to polar coordinates is beneficial because it reduces complexity by mapping a 3D surface into a 2D plane.

• In this case, the parametrized surface is represented as \(\mathbf{r}(r, \theta) = \langle r \cos(\theta), r \sin(\theta), r^2 \rangle\).
• The parameter limits are \(0 \leq r \leq 2\) and \(0 \leq \theta \leq 2\pi\).

This parametrization helps us move from a spatial cylinder into manageable slices, defined by the variables \(r\) and \(\theta\), suited for integration.

Surface Integral

Surface integrals extend the concept of line integrals to two dimensions over a surface. They are integral to Stokes' Theorem, helping to determine how a vector field behaves across a surface. By calculating a surface integral, we assess not only the flow across the boundary of the surface, but also the cumulative effect over the entire surface.

• For our exercise, Stokes' Theorem reformulates the line integral \(\int_C \mathbf{F} \cdot d \mathbf{r}\) into a surface integral: \(\iint_S (abla \times \mathbf{F}) \cdot d \mathbf{S}\).
• The notation \(d \mathbf{S}\) represents a small segment of the surface area, often a "normal" vector to the surface in its function space.

The problem converts a complex boundary line integral into a more straightforward surface integral over the disc defined by the paraboloid in polar coordinates. This approach for assessing a vector field’s influence over a surface illustrates the beauty of vector calculus applications.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields; it extends calculus from functions to vector fields. It's crucial for understanding physical phenomena where quantities have both magnitude and direction, like electromagnetic fields and fluid dynamics. Key operations in vector calculus include:

• Gradient: Measures how a scalar field changes in space.
• Divergence: Represents the rate of flux expansion or compression within a field.
• Curl: Assesses the rotation around a point in a vector field.

Stokes' Theorem, a pivotal concept in vector calculus, connects a surface integral (over the curl of a field) with a line integral (around the boundary of the surface). In practical terms, these theorems offer profound insights into natural laws—providing tools for predicting how fields behave and interact in multi-dimensional spaces.