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Magazine Chapter 19: Problem 62
Let \(\vec{H}=\left(e^{x y}+3 z+5\right) \vec{i}+\left(e^{x y}+5 z+3\right) \vec{j}+\left(3 z+e^{x y}\right) \vec{k}\) Calculate the flux of \(\overrightarrow{\boldsymbol{H}}\) through the square of side 2 with one vertex at the origin, one edge along the positive \(y\) -axis, one edge in the \(x z\) plane with \(x>0, z>0\) and the normal \(\vec{n}=\vec{i}-\vec{k}\)
Short Answer
Expert verified
The flux is approximately 0.5\(e^4 + 31\).
Step by step solution
01
Express Surface Parameterization
Given that one edge of the square is along the y-axis, we can parameterize the surface using parameters \( y \) and \( z \). Set \( x = 2 \) since the surface resides in the \( xz \)-plane and parallel to the \( xy \)-plane, and the normal is along \( \vec{i} - \vec{k} \).Thus, the parameterization is \( \vec{r}(y, z) = 2\vec{i} + y\vec{j} + z \vec{k} \), where \( 0 \leq y \leq 2 \) and \( 0 \leq z \leq 2 \).
02
Calculate Surface Normal Vector
The given normal is \( \vec{n} = \vec{i} - \vec{k} \). However, to check consistency, compute \( \frac{\partial \vec{r}}{\partial y} \times \frac{\partial \vec{r}}{\partial z} \) to find the actual normal vector associated with the parameterization:- \( \frac{\partial \vec{r}}{\partial y} = \vec{j} \)- \( \frac{\partial \vec{r}}{\partial z} = \vec{k} \)The cross-product \( \vec{j} \times \vec{k} = \vec{i} \) confirms that the orientation \( \vec{n} = \vec{i} \) indeed aligns with \( \vec{i} - \vec{k} \).
03
Compute Dot Product of \( \vec{H} \) and Normal
Evaluate \( \vec{H} \cdot \vec{n} = (e^{xy} + 3z + 5)\vec{i} \cdot \vec{i} = e^{xy} + 3z + 5 \). Substitute \( x = 2 \) throughout the square, leading to the expression becoming: \[ \vec{H} \cdot \vec{n} = e^{2y} + 3z + 5 \]
04
Calculate the Double Integral for Flux
Set up the double integral over the square to find the flux: \[ \int_{0}^{2} \int_{0}^{2} (e^{2y} + 3z + 5) \ dy \ dz \]
05
Integrate with Respect to \( y \)
Perform the integration relative to \( y \):\[ \int_{0}^{2} \left[ \left( \frac{1}{2}e^{2y} + 5y \right) \bigg|_{0}^{2} \right] dz \]Evaluating at the limits gives:\[ \int_{0}^{2} \left( \frac{1}{2}(e^4 - 1) + 10 \right) dz \]
06
Integrate with Respect to \( z \)
Now integrate over \( z \):\[ \left[ \left( \frac{1}{2}(e^4 - 1) + 10 \right)z + \frac{3}{2}z^2 \right] \bigg|_{0}^{2} = 2 \left( \frac{1}{2}(e^4 - 1) + 10 + 6 \right) \]Calculating gives:\[ = 2 \left( \frac{1}{2}e^4 - \frac{1}{2} + 16 \right) = 2 \left( \frac{1}{2}e^4 + 15.5 \right) = 0.5e^4 + 31 \]
07
Evaluate and Simplify Result
Apply any needed extra simplifications or approximations (none required here at this stage). The end result is the flux:\[ \phi = 0.5e^4 + 31 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Fields
In multivariable calculus, a vector field is a function that assigns a vector to each point in space. This can be understood as a sort of "invisible field," where at every point in space, there is a vector giving direction and magnitude. These vectors can represent various phenomena, such as force or velocity.
For example, think of the wind: at each point in the sky, there is a wind speed and direction, which can be described by a vector field. In our exercise, the vector field is expressed as \( \vec{H} = \left(e^{xy} + 3z + 5\right) \vec{i} + \left(e^{xy} + 5z + 3\right) \vec{j} + \left(3z + e^{xy}\right) \vec{k} \).
This indicates that at each point \((x, y, z)\), there is a vector defined by these components, helping us understand how quantities change across space.
For example, think of the wind: at each point in the sky, there is a wind speed and direction, which can be described by a vector field. In our exercise, the vector field is expressed as \( \vec{H} = \left(e^{xy} + 3z + 5\right) \vec{i} + \left(e^{xy} + 5z + 3\right) \vec{j} + \left(3z + e^{xy}\right) \vec{k} \).
This indicates that at each point \((x, y, z)\), there is a vector defined by these components, helping us understand how quantities change across space.
Surface Integrals
Surface integrals are a way to extend the concept of integration to over-determining areas or surfaces in space. Just as the integral in single-variable calculus calculates the area underneath a curve, the surface integral calculates a sort of 'flux' through a surface.
The concept of a surface integral is useful when determining the total flow (flux) of a field through a given surface.
In our problem, to find the flux of vector field \( \vec{H} \) through the square, we set up a surface integral using the dot product of the vector field and the surface normal vector.
The concept of a surface integral is useful when determining the total flow (flux) of a field through a given surface.
In our problem, to find the flux of vector field \( \vec{H} \) through the square, we set up a surface integral using the dot product of the vector field and the surface normal vector.
- A surface integral sums up the flow at each infinitesimally small piece of the surface.
- The result gives us the total amount of the vector field passing through the surface.
Parameterization of Surfaces
Parameterization is a technique used to define a surface using one or more parameters alongside known mathematical functions. This allows us to describe surfaces in a way that is suitable for calculus computations.
For the specified example, wherein the surface is a square, we use parameters \( y \) and \( z \) to describe the surface.
For the specified example, wherein the surface is a square, we use parameters \( y \) and \( z \) to describe the surface.
- The parameterization becomes \( \vec{r}(y, z) = 2 \vec{i} + y \vec{j} + z \vec{k} \), where \( 0 \leq y \leq 2 \) and \( 0 \leq z \leq 2 \).
- This description "maps" the surface onto a plane expressed in terms of the parameters.
Multivariable Calculus
Multivariable calculus generalizes concepts from single-variable calculus to functions of several variables. This includes operations such as differentiation and integration within higher dimensions.
Some of the key extensions include:
With the vector field \( \vec{H} \) and the parameterized surface, we utilize the operations of multivariable calculus to compute the flux through a specified region, expanding the simple idea of summing under a curve to summing the flow across a surface. In essence, multivariable calculus provides the tools we use to solve and understand these more intricate spatial analysis problems.
Some of the key extensions include:
- Handling vector functions which dictate how vectors change in space.
- Integration over curves and surfaces, which is essential in physics and engineering applications.
With the vector field \( \vec{H} \) and the parameterized surface, we utilize the operations of multivariable calculus to compute the flux through a specified region, expanding the simple idea of summing under a curve to summing the flow across a surface. In essence, multivariable calculus provides the tools we use to solve and understand these more intricate spatial analysis problems.
