Problem 13 - Calculus Early Transcendentals

Question

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) in the direction of positive orientation. \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+\mathbf{k} ; \sigma\) is the portion of the paraboloid $$ \mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+\left(1-u^{2}\right) \mathbf{k} $$ with \(1 \leq u \leq 2,0 \leq v \leq 2 \pi\)

Step-by-Step Solution

Answer

The flux of the vector field across the surface is \( 18 \pi \).

Step 1: Parameterize the Surface

The surface \( \sigma \) is given by the parameterization \( \mathbf{r}(u, v) = u \cos v \mathbf{i} + u \sin v \mathbf{j} + (1 - u^2) \mathbf{k} \), with parameters \( 1 \leq u \leq 2 \) and \( 0 \leq v \leq 2 \pi \). We need to calculate the surface integral of \( \mathbf{F} \) across this parameterized surface.

Step 2: Compute Partial Derivatives of the Parameterization

Find the partial derivatives of \( \mathbf{r}(u, v) \) with respect to \( u \) and \( v \):\[\frac{\partial \mathbf{r}}{\partial u} = \cos v \mathbf{i} + \sin v \mathbf{j} - 2u \mathbf{k},\]\[\frac{\partial \mathbf{r}}{\partial v} = -u \sin v \mathbf{i} + u \cos v \mathbf{j}.\]

Step 3: Calculate the Normal Vector

The normal vector \( \mathbf{N} \) is given by the cross product:\[\mathbf{N} = \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\cos v & \sin v & -2u \-u \sin v & u \cos v & 0\end{vmatrix} = (2u^2 \cos v) \mathbf{i} + (2u^2 \sin v) \mathbf{j} + u \mathbf{k}.\]

Step 4: Compute the Dot Product \( \mathbf{F}\cdot\mathbf{N} \)

The vector field \( \mathbf{F}(x, y, z) = x \mathbf{i} + y \mathbf{j} + \mathbf{k} \) evaluated on the surface is \( u \cos v \mathbf{i} + u \sin v \mathbf{j} + \mathbf{k} \). Compute the dot product:\[\mathbf{F} \cdot \mathbf{N} = (u \cos v \mathbf{i} + u \sin v \mathbf{j} + \mathbf{k}) \cdot (2u^2 \cos v \mathbf{i} + 2u^2 \sin v \mathbf{j} + u \mathbf{k}) = 2u^3 + u.\]

Step 5: Set Up the Surface Integral

The surface integral for flux is given by:\[\iint_{\sigma} \mathbf{F} \cdot \mathbf{N} \, dS = \iint_{D} (2u^3 + u) \, du \, dv,\]where \( D \) is the region \( 1 \leq u \leq 2, \, 0 \leq v \leq 2\pi \).

Step 6: Evaluate the Integral

Now integrate over \( u \) from 1 to 2 and \( v \) from 0 to \( 2 \pi \):\[\int_{0}^{2\pi} \int_{1}^{2} (2u^3 + u) \, du \, dv.\]First, integrate with respect to \( u \):\[\int_{1}^{2} (2u^3 + u) \, du = \left[ \frac{1}{2}u^4 + \frac{1}{2}u^2 \right]_{1}^{2} = \left[ \frac{1}{2}(16 - 1) + \frac{1}{2}(4 - 1) \right] = 15/2 + 3/2 = 9.\]Then integrate with respect to \( v \):\[\int_{0}^{2\pi} 9 \, dv = 9 \cdot 2\pi = 18 \pi.\]

Step 7: Final Result

The flux of the vector field \( \mathbf{F} \) across \( \sigma \) is \( 18 \pi \). The positive orientation is correctly taken care of by directly using the cross product in our calculation of the normal vector.

Key Concepts

Flux

Flux is a fundamental concept in vector calculus that measures the flow of a vector field through a surface. Imagine water flowing through a net—flux would be the amount of water passing through the net. In mathematics, we calculate flux to understand how much of a field (like a magnetic or electric field) passes through a given surface.
When computing the flux of a vector field \( \mathbf{F} \) across a surface \( \sigma \), we use the surface integral \( \iint_{\sigma} \mathbf{F} \cdot \mathbf{N} \, dS \).

• \( \mathbf{N} \) represents the normal vector to the surface \( \sigma \).
• \( \, dS \) denotes the differential area element of the surface.

The goal is to take into account the direction and magnitude of the vector field on the surface. The dot product \( \mathbf{F} \cdot \mathbf{N} \) projects the vector field onto the normal vector, ensuring only the component of the field passing through the surface is considered. This gives us a scalar field that we integrate over the surface to determine the total flux.

Vector Calculus

Vector calculus is a branch of mathematics focusing on vector fields and operations such as differentiation and integration across those fields. It allows us to analyze fields that vary in space and are sensitive to direction, like gravitational, electric, and magnetic fields.
For instance, in vector calculus, we work with the gradient, divergence, and curl, which are fundamental operators. These operators help us understand:

• How a field changes from point to point (gradient).
• How a field spreads out or contracts (divergence).
• The rotational behavior at any point in the field (curl).

In the context of surface integrals, vector calculus provides tools to compute the flux of a vector field across a surface, which is essential for understanding how vector fields interact with surfaces in three dimensions. Using parameterizations and normal vectors, vector calculus helps set up and solve these problems efficiently.

Parameterization

Parameterization involves expressing a surface in terms of two parameters, which are typically written as \( u \) and \( v \). For a surface in three-dimensional space, a parameterization \( \mathbf{r}(u, v) \) describes a mapping from the parameter space to every point on the surface.
In our exercise, the surface \( \sigma \) is parameterized by \[ \mathbf{r}(u, v) = u \cos v \mathbf{i} + u \sin v \mathbf{j} + (1 - u^2) \mathbf{k} \].
This expression tells us:

• The first two components, \( u \cos v \) and \( u \sin v \), allow us to navigate the surface in a cylindrical fashion around an axis.
• The third component, \( (1 - u^2) \), brings a vertical deformation, creating a paraboloid shape.

Parameterization simplifies working with complex surfaces by turning them into equations of familiar shapes, making it easier to apply calculus and compute quantities like flux.

Normal Vector

The normal vector is essential in many calculations involving surfaces, especially surface integrals. It is a vector perpendicular to a given surface at a point. The direction of this normal vector is significant for calculations like flux, as it defines the positive direction for measuring the flow through a surface.
For a parameterized surface \( \mathbf{r}(u, v) \), the normal vector \( \mathbf{N} \) can be found by taking the cross product of the partial derivatives of \( \mathbf{r} \) with respect to its parameters.
In the example provided, the normal vector is calculated as:
\[ \mathbf{N} = \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} = (2u^2 \cos v) \mathbf{i} + (2u^2 \sin v) \mathbf{j} + u \mathbf{k} \].
Finding the normal vector enables the calculation of surface integrals by aligning the flux with the surface's orientation. It helps ensure that we measure the correct amount of the vector field passing through the surface, adhering to its natural orientation.