Problem 15 - Advanced Engineering Mathematics

Question

In Problems 13-16, use Stokes' theorem to evaluate \(\iint_{S}\) (curl \(\left.\mathbf{F}\right) \cdot \mathbf{n} d S\). Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=3 x^{2} \mathbf{i}+8 x^{3} y \mathbf{j}+3 x^{2} y \mathbf{k} ; S\) that portion of the plane \(z=x\) that lies inside the rectangular cylinder defined by the planes \(x=0, y=0, x=2, y=2\)

Step-by-Step Solution

Answer

Apply Stokes' theorem: the result is obtained by evaluating the line integral along the boundary C.

Step 1: Background of Stokes' Theorem

Stokes' Theorem states that \(\iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r}\), where \(C\) is the positively oriented boundary curve of the surface \(S\) and \(\mathbf{n}\) is the outward normal to \(S\).

Step 2: Compute the Curl of \(\mathbf{F}\)

The vector field \(\mathbf{F}\) is given by \(\mathbf{F} = 3x^{2} \mathbf{i} + 8x^{3}y \mathbf{j} + 3x^{2}y \mathbf{k}\). Compute the curl: \(abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ 3x^2 & 8x^3y & 3x^2y \end{vmatrix}\). "},{

Step 3: Find the Boundary Curve C

The boundary curve \(C\) is the intersection of the plane \(z=x\) with the rectangular cylinder defined by \(0\leq x \leq 2\), \(0\leq y \leq 2\). Thus, \(C\) is a rectangle in the xy-plane, connecting the points (0,0), (2,0), (2,2), and (0,2), since the entire surface is at height \(z=x\).

Step 4: Parametrize the Boundary Curve C

We parametrize \(C\) by using the edges of the rectangle.1. From \((0,0)\) to \((2,0)\): \(\mathbf{r}_{1}(t) = (t, 0, t)\), \(0 \leq t \leq 2\).2. From \((2,0)\) to \((2,2)\): \(\mathbf{r}_{2}(t) = (2, t, 2)\), \(0 \leq t \leq 2\).3. From \((2,2)\) to \((0,2)\): \(\mathbf{r}_{3}(t) = (2-t, 2, 2-t)\), \(0 \leq t \leq 2\).4. From \((0,2)\) to \((0,0)\): \(\mathbf{r}_{4}(t) = (0, 2-t, 0)\), \(0 \leq t \leq 2\).

Step 5: Evaluate the Line Integral \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\)

Compute the contribution to the line integral from each segment of the path:1. For \(\mathbf{r}_{1}\): \(\int_{0}^{2} \mathbf{F}(\mathbf{r}_{1}(t)) \cdot \frac{d\mathbf{r}_{1}}{dt} \, dt\).2. For \(\mathbf{r}_{2}\): \(\int_{0}^{2} \mathbf{F}(\mathbf{r}_{2}(t)) \cdot \frac{d\mathbf{r}_{2}}{dt} \, dt\).3. For \(\mathbf{r}_{3}\): \(\int_{0}^{2} \mathbf{F}(\mathbf{r}_{3}(t)) \cdot \frac{d\mathbf{r}_{3}}{dt} \, dt\).4. For \(\mathbf{r}_{4}\): \(\int_{0}^{2} \mathbf{F}(\mathbf{r}_{4}(t)) \cdot \frac{d\mathbf{r}_{4}}{dt} \, dt\).Compute each integral separately and sum them.

Step 6: Apply Stokes' Theorem Conclusion

Since the line integrals around the closed curveC are evaluated, the sum of these line integrals will be the result of \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\), which equals the surface integral \(\iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \) by Stokes' theorem.

Key Concepts

Vector Calculus

Vector Calculus is an essential branch of mathematics that deals with vector fields and operations involving vectors, primarily in three-dimensional space. It focuses on differentiating and integrating vector functions and is a crucial tool in physics and engineering.

In vector calculus, we often deal with concepts such as divergence, gradient, and curl — each has different applications and interpretations. These operations allow us to study the behavior of vector fields, like how things change or flow within a field.

• Divergence: Measures the magnitude of a source or sink at a given point in a vector field.
• Gradient: Indicates the direction and rate of the fastest increase of a scalar field.
• Curl: Describes the rotation or "twist" of a vector field.

In the context of Stokes' Theorem, vector calculus provides the framework for analyzing how the rotation of a field (curl) across a surface relates to its circulation along a curve.

Curl of a Vector Field

The Curl of a Vector Field is an operation that measures a field's rotation or "twist" at each point. To compute it, we apply the curl operator, denoted by \( abla \times \mathbf{F} \), to a vector field \( \mathbf{F} \). The result is another vector field.

Mathematically, for a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is computed using the determinant:\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \]This expression gives a new vector field that describes how \( \mathbf{F} \) rotatesThis is crucial in fluid dynamics and electromagnetism, where the curl represents rotational motion or the magnetic field induced by a changing electric field.
Understanding curl is essential when applying Stokes' Theorem, as it is part of the theorem's core relations.

Line Integrals

A Line Integral is a type of integral where a function is integrated along a path or curve in a vector field. It is useful for finding work done by a force along a path, or the flow of a fluid along a curve.

When calculating line integrals, the path taken by the curve greatly influences the result. For a vector field \( \mathbf{F} \), the line integral along a curve \( C \) is written as:\[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} \]where \( d\mathbf{r} \) is a small vector tangent to the curve \( C \).

Line integrals follow distinct paths:

• If \( \mathbf{F} \) is a force field, the line integral can represent the work done on a particle moving along \( C \).
• If \( \mathbf{F} \) is a velocity field, it could describe the flow of fluid along \( C \).

In the context of Stokes' Theorem, evaluating the line integral around the closed curve allows us to relate it to the surface integral of the curl over the surface the curve bounds.

Parametrization of Curves

Parametrization of Curves involves expressing a curve using a parameter, typically \( t \), to describe each point along the curve in terms of one or more equations. This method simplifies the calculation of line integrals and other operations.

For example, consider a path defined in three dimensions:

• A curve from \( (x_0, y_0, z_0) \) to \( (x_1, y_1, z_1) \) could be written as:
  \( \mathbf{r}(t) = (1-t)(x_0, y_0, z_0) + t(x_1, y_1, z_1) \) for \( 0 \leq t \leq 1 \)

This vector function \( \mathbf{r}(t) \) traces out the curve as \( t \) varies from 0 to 1. The derivatives of these parameterizations provide the tangent vectors required for line integrals.

Using parameterizations is particularly important when dealing with piecewise curves, enabling us to integrate across each segment of the path individually.
In our problem, we break the boundary curve \( C \) into segments and parameterize each to compute the line integrals efficiently, which are then used in Stokes' Theorem.