Problem 363 - Calculus Volume 3

Question

[T] Use a CAS to evaluate \(\iint_{S} \operatorname{curl}(\mathbf{F}) \cdot d \mathbf{S},\) where \(\mathbf{F}(x, y, z)=2 z \mathbf{i}+3 x \mathbf{j}+5 y \mathbf{k}\) and \(S\) is the surface parametrically \(\mathbf{r}(r, \theta)=r \cos \theta \mathbf{i}+r \sin \theta \mathbf{j}+\left(4-r^{2}\right) \mathbf{k}\) \((0 \leq \theta \leq 2 \pi, 0 \leq r \leq 3)\)

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Calculate Curl of F

The first step is to calculate the curl of the vector field \( \mathbf{F} = 2z \mathbf{i} + 3x \mathbf{j} + 5y \mathbf{k} \). The curl is given by:\[abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k}\]Substituting the components, \( F_x = 2z \), \( F_y = 3x \), \( F_z = 5y \), we get:* \( \frac{\partial F_z}{\partial y} = 5 \) and \( \frac{\partial F_y}{\partial z} = 0 \).* \( \frac{\partial F_x}{\partial z} = 2 \) and \( \frac{\partial F_z}{\partial x} = 0 \).* \( \frac{\partial F_y}{\partial x} = 3 \) and \( \frac{\partial F_x}{\partial y} = 0 \).Thus, \( abla \times \mathbf{F} = 5\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \).

Step 2: Parameterize the Surface

The surface \( S \) is parameterized by \( \mathbf{r}(r, \theta) = r \cos\theta \mathbf{i} + r \sin\theta \mathbf{j} + (4 - r^2) \mathbf{k} \). The parameters are given as \( 0 \leq \theta \leq 2\pi \) and \( 0 \leq r \leq 3 \).

Step 3: Compute dS with the Cross Product of Partial Derivatives

The differential surface element \( d\mathbf{S} \) is computed using:\[ d\mathbf{S} = \left( \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} \right) dr d\theta \]Calculate partial derivatives:* \( \frac{\partial \mathbf{r}}{\partial r} = \cos\theta \mathbf{i} + \sin\theta \mathbf{j} - 2r \mathbf{k} \)* \( \frac{\partial \mathbf{r}}{\partial \theta} = -r\sin\theta \mathbf{i} + r\cos\theta \mathbf{j} \)Compute the cross product:\[ \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \cos\theta & \sin\theta & -2r \ -r\sin\theta & r\cos\theta & 0 \end{vmatrix} \]Evaluating the determinant, we get:\[ \mathbf{i}(2r^2 \cos\theta) + \mathbf{j}(2r^2 \sin\theta) + \mathbf{k}(r) \]

Step 4: Evaluate the Surface Integral Using a CAS

The surface integral becomes:\[ \iint_{S} abla \times \mathbf{F} \cdot d\mathbf{S} = \int_0^{2\pi} \int_0^3 (5 \cdot 2r^2 \cos\theta + 2 \cdot 2r^2 \sin\theta + 3 \cdot r) dr d\theta \]Simplifying the integrand:\[ = \int_0^{2\pi} \int_0^3 (10r^2 \cos\theta + 4r^2 \sin\theta + 3r) dr d\theta \]Using a CAS, the evaluated integral provides the result of the surface integral.

Step 5: Final Result

The computed result evaluates to zero. Therefore, the given line integral evaluates to:\( \iint_{S} abla \times \mathbf{F} \cdot d\mathbf{S} = 0 \).

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and differentiates and integrates vector-valued functions. It's crucial in understanding physical fields like electromagnetism and fluid dynamics. In our task, the main goal of vector calculus was to compute the curl of a vector field, which represents the field's rotation. The curl is a vector itself and is obtained using the cross-product of the del operator with the given vector field.

In our example,

• The vector field is \( \mathbf{F} = 2z \mathbf{i} + 3x \mathbf{j} + 5y \mathbf{k} \).
• Finding the curl involves partial derivatives like \( abla \times \mathbf{F} = \frac{\partial F_z}{\partial y} \mathbf{i} + \frac{\partial F_x}{\partial z} \mathbf{j} + \frac{\partial F_y}{\partial x} \mathbf{k} \).

This operation helps identify the presence of any rotational behavior in the field and lays the foundation for evaluating the integral over a surface.

Parametric Surfaces

In vector calculus, parametric surfaces represent a crucial concept. These surfaces are described using parameters, often denoted by \( (r, \theta) \), to express every point on the surface. This type of parameterization enables us to handle complex surfaces in a mathematic framework.

For the surface \( S \) given by:

• \( \mathbf{r}(r, \theta) = r \cos\theta \mathbf{i} + r \sin\theta \mathbf{j} + (4 - r^2) \mathbf{k} \)

parametrization lets us describe points on a paraboloid surface in cylindrical coordinates.

Using parametric equations simplifies the process of integrating over complex surfaces, translating the integral into one over the specified range of parameters \( r \) and \( \theta \). It effectively converts the surface geometry into manageable calculus operations.

Curl of a Vector Field

The curl of a vector field measures the tendency of the field to rotate around a point. It's a fundamental operation in vector calculus and is key for analyzing vector fields representing fluid flow or electromagnetic fields. To find the curl, we use the formula:

\[abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k}\]
For the vector field \( \mathbf{F}(x, y, z) \) in our example:

• \( F_x = 2z \), \( F_y = 3x \), and \( F_z = 5y \).

The calculated curl \( abla \times \mathbf{F} = 5\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) explains how aspects of the vector field vary and rotate, providing insight into its underlying structure.

Double Integrals

Double integrals extend the concept of integration to functions of two variables. They allow for the evaluation of surface area or volume under a given surface, considering both dimensions. When involving vector fields, double integrals help us find quantities like flux or in this case, the surface integral over a parametric surface.

For the given surface in the exercise, calculating \( d\mathbf{S} \) involved finding the cross product of partial derivatives:

• \( \frac{\partial \mathbf{r}}{\partial r} \) and \( \frac{\partial \mathbf{r}}{\partial \theta} \) were obtained to describe surface changes.

This enabled us to form a double integral over \( r \) and \( \theta \). By integrating the dot product of the curl of \( \mathbf{F} \) with \( d\mathbf{S} \), we determined the net rotation effect of \( \mathbf{F} \) over the surface \( S \).

The use of a computer algebra system (CAS) greatly facilitated solving the integral, especially when complex functions are involved, thus streamlining the entire computation process.