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Magazine Chapter 16: Problem 28
Evaluate the surface integral \( \displaystyle \iint_S \textbf{F} \cdot d\textbf{S} \) for the given vector field \( \textbf{F} \) and the oriented surface \( S \). In other words, find the flux of \( \textbf{F} \) across \( S \). For closed surfaces, use the positive (outward) orientation. \( \textbf{F}(x, y, z) = yz \, \textbf{i} + zx \, \textbf{j} + xy \, \textbf{k} \), \( S \) is the surface \( z = x \sin y \), \( 0 \leqslant x \leqslant 2 \), \( 0 \leqslant y \leqslant \pi \), with upward orientation
Short Answer
Expert verified
The flux of \( \textbf{F} \) across \( S \) is \( 2\pi \).
Step by step solution
01
Parametrize the Surface
The surface given is described by the equation \( z = x \sin y \). We can introduce parameters \( u \) and \( v \) to parametrize the surface as \( \textbf{r}(u,v) = u\, \textbf{i} + v\, \textbf{j} + u \sin v\, \textbf{k} \), where \( 0 \leqslant u \leqslant 2 \) and \( 0 \leqslant v \leqslant \pi \).
02
Compute the Partial Derivatives
Calculate the partial derivatives of \( \textbf{r}(u,v) \) to find the tangent vectors:\[ \textbf{r}_u = \frac{\partial \textbf{r}}{\partial u} = \textbf{i} + \sin v \textbf{k} \]\[ \textbf{r}_v = \frac{\partial \textbf{r}}{\partial v} = \textbf{j} + u \cos v \textbf{k} \].
03
Calculate the Normal Vector
Find the normal vector, \( \textbf{N} \), using the cross product of \( \textbf{r}_u \) and \( \textbf{r}_v \):\[ \textbf{N} = \textbf{r}_u \times \textbf{r}_v = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ 1 & 0 & \sin v \ 0 & 1 & u \cos v \end{vmatrix} = -\sin v \, \textbf{i} - u \cos v \, \textbf{j} + \textbf{k} \]Ensure the normal vector points upwards. In this case, the \( \textbf{k} \) component is positive, confirming the upward orientation.
04
Evaluate \( \textbf{F} \cdot \textbf{N} \)
Substitute the parametric equations into the vector field \( \textbf{F} \):\[ \textbf{F}(u,v) = (v u \sin v) \textbf{i} + (u^2 \sin v) \textbf{j} + (uv) \textbf{k} \].Now compute \( \textbf{F} \cdot \textbf{N} \):\[ \textbf{F}(u,v) \cdot \textbf{N} = (v u \sin v)(-\sin v) + (u^2 \sin v)(-u \cos v) + (uv) \cdot 1 \],simplifying to \[ -v u \sin^2 v - u^3 \sin v \cos v + uv \].
05
Setup and Solve Integral
Evaluate the surface integral of \( \textbf{F} \cdot \textbf{N} \) over the domain. Set up the integral:\[ \iint_S \textbf{F} \cdot \textbf{N} \, dS = \int_{0}^{\pi} \int_{0}^{2} (-v u \sin^2 v - u^3 \sin v \cos v + uv) \, du \, dv \].Calculate the inner integral first with respect to \( u \), followed by the outer integral with respect to \( v \), evaluating each section by section.
06
Evaluate Inner Integral
Evaluate the integral with respect to \( u \):\( \int_{0}^{2} (-v u\sin^2 v - u^3\sin v\cos v + uv) \, du \)- Integrate term by term: 1. \( -v \sin^2 v \int u \, du = -v \sin^2 v \left[\frac{u^2}{2}\right]_{0}^{2} = -2v \sin^2 v \) 2. \( -\sin v\cos v \int u^3 \, du = -\sin v\cos v \left[\frac{u^4}{4}\right]_{0}^{2} = -4\sin v\cos v \) 3. \( v \int u \, du = v \left[\frac{u^2}{2}\right]_{0}^{2} = 2v \)Combine:\[ \iint_S \textbf{F} \cdot \textbf{N} \, dS = \int_{0}^{\pi} (-2v \sin^2 v - 4\sin v \cos v + 2v) \, dv \].
07
Evaluate Outer Integral
Now evaluate the integral with respect to \( v \):\( \int_{0}^{\pi} (-2v \sin^2 v - 4\sin v \cos v + 2v) \, dv = \int_{0}^{\pi} (-2v \sin^2 v) \, dv - \int_{0}^{\pi} 4\sin v \cos v \, dv + \int_{0}^{\pi} 2v \, dv \).Perform each integral separately. To solve the integral \( \int 4 \sin v \cos v \, dv \), use substitution or trigonometric identities to solve it. Evaluate each integral over \( [0, \pi] \).
08
Compute Final Answer
Sum the evaluated terms from Step 7 to find the total result of the integral. The sums of the integrals will give you the flux across the surface.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Flux Across Surface
Flux represents the quantity of a field passing through a surface. In this case, as we evaluate the surface integral \( \iint_S \textbf{F} \cdot d\textbf{S} \), we are determining how much of the vector field \( \textbf{F} \) flows through the oriented surface \( S \). The orientation described must be upward or outward for closed surfaces, ensuring the calculated flux is accurate. This involves evaluating the component of the vector field perpendicular to the surface and integrates it across the surface area. Understanding this helps us in solving real-world problems like fluid flow or electromagnetic field analysis.
Parametrization of Surfaces
To compute a surface integral, we start by expressing our surface in parametric form. For the given surface \( z = x \sin y \), we parametrize using two variables: \( u \) and \( v \). The parametrization is \( \textbf{r}(u,v) = u \, \textbf{i} + v \, \textbf{j} + u \sin v \, \textbf{k} \), ranging from \( 0 \leq u \leq 2 \) and \( 0 \leq v \leq \pi \). Parametrization simplifies complex surfaces into an expression that can easily be manipulated using calculus, allowing us to compute derivatives and integrals on those surfaces with ease.
Vector Field Integration
Vector field integration across a surface involves integrating the dot product of a vector field \( \textbf{F} \) and the surface's normal vector over the surface area. The expression \( \textbf{F}(u,v) = (v u \sin v) \textbf{i} + (u^2 \sin v) \textbf{j} + (uv) \textbf{k} \) represents the vector field transformed into our specified parameters. We evaluate the dot product \( \textbf{F} \cdot \textbf{N} \), where \( \textbf{N} \) is the normal vector of the surface, to find the integrand. Performing integration over defined limits, we handle the calculations term by term. This process finds practical applications in physics and engineering, serving to calculate quantities such as fluid flow through a surface.
Normal Vector Calculation
The normal vector is crucial in surface integrals as it dictates the direction against which the vector field is projected. We find the normal vector \( \textbf{N} \) by calculating the cross product of tangent vectors \( \textbf{r}_u \) and \( \textbf{r}_v \) derived from our parametrization. For this problem, calculating \( \textbf{N} = -\sin v \, \textbf{i} - u \cos v \, \textbf{j} + \textbf{k} \) is needed. It's important to ensure the normal vector matches the surface's orientation requirement鈥攊n this exercise, upward. The normal vector directly affects the integrand value, leading to accurate results for surface flux calculations.
