Problem 13 - 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. $$ \begin{aligned} &\mathbf{F}=6 y z \mathbf{i}+5 x \mathbf{j}+y z e^{x^{2}} \mathbf{k} ; S \text { that portion of the paraboloid }\\\ &z=\frac{1}{4} x^{2}+y^{2} \text { for } 0 \leq z \leq 4 \end{aligned} $$

Step-by-Step Solution

Answer

The result of the integral is \( -152\pi \).

Step 1: Understand Stokes' Theorem

Stokes' theorem states \( \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} \), where \(C\) is the boundary curve of the surface \(S\) and is oriented counterclockwise as seen from above the surface. Our task is to compute this line integral.

Step 2: Find the Boundary Curve C

The given surface is the paraboloid \( z = \frac{1}{4} x^2 + y^2 \) for \( 0 \leq z \leq 4 \). The boundary \( C \) is the circle where \( z = 4 \) (i.e., \( \frac{1}{4}x^2 + y^2 = 4 \)), which simplifies to \( x^2 + 4y^2 = 16 \). This is an ellipse with semi-major axis length 4 and semi-minor axis length 2.

Step 3: Parameterize the Boundary Curve C

Parameterize the boundary ellipse using \( x = 4 \cos t \) and \( y = 2 \sin t \), where \( 0 \leq t \leq 2\pi \). The position vector for the boundary curve \( \mathbf{r}(t) = \langle 4 \cos t, 2 \sin t, 4 \rangle \).

Step 4: Compute \( d\mathbf{r} \)

The derivative \( \mathbf{r}'(t) = \langle -4 \sin t, 2 \cos t, 0 \rangle \). So, \( d\mathbf{r} = \mathbf{r}'(t) \, dt = \langle -4 \sin t, 2 \cos t, 0 \rangle \, dt \).

Step 5: Evaluate \( \mathbf{F}(\mathbf{r}(t)) \)

Substitute \( x = 4 \cos t \), \( y = 2 \sin t \), and \( z = 4 \) into \( \mathbf{F} \). This gives \( \mathbf{F}(\mathbf{r}) = \langle 48 \sin t, 20 \cos t, 8 e^{16} \sin t \rangle \).

Step 6: Compute the Dot Product

Calculate \( \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) = (48 \sin t)(-4 \sin t) + (20 \cos t)(2 \cos t) + (8 e^{16} \sin t)(0) \). Simplify this expression to get \( -192 \sin^2 t + 40 \cos^2 t \).

Step 7: Solve the Integral

Integrate the dot product over \( t \) from 0 to \( 2\pi \). The integrand is \( -192 \sin^2 t + 40 \cos^2 t = -192(1 - \cos^2 t) + 40 \cos^2 t = -192 + 232 \cos^2 t \). The integral becomes \(-192(2\pi) + 232 \int_0^{2\pi} \cos^2 t \, dt \).

Step 8: Compute \( \int_0^{2\pi} \cos^2 t \, dt \)

To evaluate \( \int_0^{2\pi} \cos^2 t \, dt \), use the identity \( \cos^2 t = \frac{1 + \cos 2t}{2} \). This gives \[ \int_0^{2\pi} \cos^2 t \, dt = \int_0^{2\pi} \frac{1}{2} + \frac{\cos 2t}{2} \, dt. \].Since \( \int_0^{2\pi} \cos 2t \, dt = 0 \), the integral simplifies to \[ \int_0^{2\pi} \frac{1}{2} \, dt = \left[ \frac{t}{2} \right]_0^{2\pi} = \pi. \]

Step 9: Complete the Solution

Substitute back into the main integral: \( -192(2\pi) + 232(\pi) = -384\pi + 232\pi = -152\pi. \)

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics focusing on the analysis of vector fields. These fields are represented by vector quantities associated with each point in a space. One common application of vector calculus is in the evaluation of vector fields over surfaces using theorems like Stokes' Theorem. This theorem is integral in relating the line integrals around a closed curve to a double integral over the surface bounded by that curve. Specifically, it states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of the field over the surface it encloses.
Stokes' Theorem can be formally expressed as:
\[abla \times \mathbf{F} \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r}\]
where \(abla \times \mathbf{F}\) is the curl of the vector field \(\mathbf{F}\), \(\mathbf{n}\) is the unit normal to the surface, \(dS\) is an element of surface area, and \(C\) is the curve bounding the surface. This relationship provides a useful tool for converting difficult surface integrals into more manageable line integrals, simplifying complex problems in physics and engineering.

Line Integrals

Line integrals are a central concept in vector calculus, allowing us to integrate functions over a curve. These integrals come in two flavors: scalar line integrals and vector line integrals. In the context of vector fields, line integrals measure the work done by a force field along a path or the circulation of the field along a closed loop.
Given a vector field \(\mathbf{F}\) and a parameterized curve \(C\) defined by \( \mathbf{r}(t) = (x(t), y(t), z(t)) \), where \(t\) ranges over some interval \([a, b]\), the vector line integral is computed as:
\[\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt\]
This dot product \(\mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\) represents the component of the vector field \(\mathbf{F}\) tangential to the curve. Line integrals are essential in applying Stokes' Theorem, where they link the circulation of a vector field along the boundary of a surface to the curl of the field across the surface itself.

Parametrization of Curves

Parametrization is a technique used in calculus to express a curve as a set of equations depending on a variable, usually \(t\), known as the parameter. This method offers a flexible way to describe curves in different coordinate systems, which is especially useful when evaluating line integrals.
For example, consider an ellipse defined by the equation \(x^2 + 4y^2 = 16\). We can parameterize this ellipse by taking \(x = 4 \cos t\) and \(y = 2 \sin t\), where \(t\) ranges from \(0\) to \(2\pi\). Thus, the parametric equations become a function of \(t\):
\[\mathbf{r}(t) = \langle 4 \cos t, 2 \sin t, 4 \rangle\]
In this representation, the variable \(t\) sweeps the curve from start to finish, capturing both its geometric shape and direction. Parametrization is vital in simplifying many complex calculus problems, often transforming intimidating integrals into tractable mathematical expressions. It allows us to consider the orientation and path traced by the curve, aspects that are crucial when working with vector fields and applying theorems like Stokes'.