Problem 15 - Calculus Early Transcendentals

Question

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) in the direction of positive orientation. \(\mathbf{F}(x, y, z)=\sqrt{x^{2}+y^{2}} \mathbf{k} ; \sigma\) is the portion of the cone $$ \mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+2 u \mathbf{k} $$ with \(0 \leq u \leq \sin v, 0 \leq v \leq \pi\)

Step-by-Step Solution

Answer

The flux is \( \frac{3\sqrt{5}}{32} \pi \).

Step 1: Calculate the parameterization surface normal

The given parameterization of the surface is \( \mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+2 u \mathbf{k} \). First, find the partial derivatives with respect to \( u \) and \( v \):- \( \frac{\partial \mathbf{r}}{\partial u} = \cos v \mathbf{i} + \sin v \mathbf{j} + 2 \mathbf{k} \)- \( \frac{\partial \mathbf{r}}{\partial v} = -u \sin v \mathbf{i} + u \cos v \mathbf{j} \).Then, find the cross product to get the normal vector:\[\mathbf{n} = \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \cos v & \sin v & 2 \ -u \sin v & u \cos v & 0 \end{vmatrix}\].

Step 2: Solve the determinant for the normal vector

Evaluate the determinant:\[\mathbf{n} = \mathbf{i}(0 - 2u \cos v) - \mathbf{j}(0 + 2u \sin v) + \mathbf{k}(u \cos^2 v + u \sin^2 v)\]Simplify using \( \cos^2 v + \sin^2 v = 1 \), then:\(\mathbf{n} = -2u \cos v \mathbf{i} - 2u \sin v \mathbf{j} + u \mathbf{k}\).

Step 3: Compute the dot product \( \mathbf{F} \cdot \mathbf{n} \)

Given \( \mathbf{F}(x, y, z) = \sqrt{x^2 + y^2} \mathbf{k} \), where \( x = u \cos v \), \( y = u \sin v \), and the cone's height is \( z = 2u \). Then, the dot product \( \mathbf{F} \cdot \mathbf{n} \) is:\[\mathbf{F} \cdot \mathbf{n} = \sqrt{(u \cos v)^2 + (u \sin v)^2} \cdot u = u^2\].

Step 4: Set up and calculate the flux integral

The flux across the surface \( \sigma \) is an integral of the dot product \( \mathbf{F} \cdot \mathbf{n} \) over the region. Evaluate:\[\iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS = \int_0^\pi \int_0^{\sin v} u^2 \, dA\]where \( dA = |\mathbf{n}| \, du \, dv \).Find the magnitude of the normal vector:\[|\mathbf{n}| = \sqrt{(-2u \cos v)^2 + (-2u \sin v)^2 + u^2} = \sqrt{5u^2} = \sqrt{5}u\].

Step 5: Evaluate the double integral

The area element becomes \( dA = \sqrt{5}u \, du \, dv \).Substitute this into the integral:\[= \int_0^\pi \int_0^{\sin v} u^3 \sqrt{5} \, du \, dv\]First, integrate with respect to \( u \):\[\int_0^{\sin v} u^3 \sqrt{5} \, du = \frac{\sqrt{5}}{4} [u^4]_0^{\sin v} = \frac{\sqrt{5}}{4} (\sin v)^4\]Now integrate with respect to \( v \):\[\int_0^\pi \frac{\sqrt{5}}{4} (\sin v)^4 \, dv = \frac{\sqrt{5}}{4} \left(\int_0^\pi (\sin v)^4 \, dv\right)\].

Step 6: Calculate the integral of \( (\sin v)^4 \)

To integrate \((\sin v)^4\), we can use the trigonometric identity:\[(\sin v)^4 = \frac{3}{8} - \frac{1}{4}\cos(2v) + \frac{1}{8}\cos(4v)\]Evaluate:\[\int_0^\pi (\sin v)^4 \, dv = \int_0^\pi \left(\frac{3}{8} - \frac{1}{4}\cos(2v) + \frac{1}{8}\cos(4v)\right) dv\]With symmetry and intervals, evaluate these integrals to obtain:\[\frac{3}{8}\pi\].

Step 7: Finalize your answer

Substituting back into the complete flux integral:\[\frac{\sqrt{5}}{4} \cdot \frac{3}{8}\pi = \frac{3\sqrt{5}}{32}\pi\].Thus, the flux of the vector field \( \mathbf{F} \) across the surface \( \sigma \) is \( \frac{3\sqrt{5}}{32}\pi \).

Key Concepts

Vector Fields

Vector fields are mathematical tools used to represent spatially varying quantities, such as fluid flow or electromagnetic force fields. They assign a vector to every point in space. In our exercise, the vector field is given by \( \mathbf{F}(x, y, z) = \sqrt{x^2 + y^2} \mathbf{k} \). A vector in this field has a magnitude directly related to the distance from the z-axis in the xy-plane and is directed upwards along the z-axis.

Vector fields are crucial in many physical applications, helping to visualize and calculate quantities like mass flow, heat distribution, or magnetic force in a region. Understanding vector fields involves recognizing how the direction and magnitude vary throughout space, which fundamentally leads us into using techniques like surface integrals to calculate important physical quantities like flux.

When engaging with vector fields, it's useful to:

• Identify the components of the field.
• Understand how they vary in the space of interest.
• Consider the physical problem being modeled by the field.

Surface Integrals

Surface integrals are an extension of multiple integrals and are used to calculate the total of some quantity over a curved surface in three-dimensional space. They are fundamental in fields like electromagnetism and fluid dynamics, as they help to compute fluxes and other surface-related properties.

In our exercise, we use a surface integral to calculate the flux of a vector field through a specified surface. This involves evaluating the integral \( \iint_{\sigma} \mathbf{F} \cdot \mathbf{n} \, dS \), which combines the vector field \( \mathbf{F} \) with a unit normal vector \( \mathbf{n} \) to the surface. The resulting scalar field \( \mathbf{F} \cdot \mathbf{n} \) is integrated over the surface \( \sigma \).

Key steps in solving surface integrals include:

• Determining the parameterization of the surface.
• Calculating the normal vector through cross products.
• Finding the magnitude of the normal vector for the differential area element \( dA \).
• Setting up and resolving the integral limits according to the specified parameterizations.

Cone Parameterization

Parameterizing a cone involves expressing the coordinates of points on the cone's surface in terms of two parameters. The parameterization \( \mathbf{r}(u, v) = u \cos v \mathbf{i} + u \sin v \mathbf{j} + 2u \mathbf{k} \) is used to describe the surface of the cone in our example. Here, \( u \) and \( v \) serve as parameters constrained by specific bounds that define the portion of the cone being studied.

The role of parameterization in vector calculus problems is to transform potentially complex shapes into forms easier to handle mathematically. This simplifies problem calculation by reducing the dimensionality of the integration process and aligning with coordinate systems that make geometric intuition more straightforward.

When you parameterize a cone:

• Choose parameters that naturally fit the symmetry or geometry of the cone.
• Ensuring that the parameter bounds encapsulate the desired portion of the surface.
• Understand how these parameters influence the spatial interpretation of the surface.

Vector Calculus

Vector calculus is a branch of mathematics focused on vector-valued functions and differential operators. It combines the principles of calculus with vector algebra and is foundational in modern physical theories and engineering.

In our exercise, vector calculus allows us to handle vector fields and surface integrations effectively, utilizing operations like dot products and cross products to find values necessary for practical predictions and analyses. The study incorporates methods to differentiate and integrate vector fields, enabling calculations of flux, circulation, and other dynamic properties associated with field behavior.

Core operations and concepts in vector calculus include:

• Gradient, divergence, and curl - operations that reveal properties of scalar and vector fields.
• Line integrals - integration over curves, often used to determine work done by a force field.
• Surface integrals - integration over a surface, providing measures like flux.
• Stokes's Theorem and Gauss's Divergence Theorem, connecting surface and volume integrals, essential for simplifying complex calculations.

Understanding these basics helps in creating bridges between abstract mathematical theories and tangible physical phenomena.