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Magazine Chapter 19: Problem 3
Compute the flux integral \(\int_{S} \vec{F} \cdot d \vec{A}\) in two ways, if possible, directly and using the Divergence Theorem. In each case, \(S\) is closed and oriented outward. \(\vec{F}=x^{2} \vec{i}+2 y^{2} \vec{j}+3 z^{2} \vec{k}\) and \(S\) is the surface of the box with faces \(x=1, x=2, y=0, y=1, z=0, z=1\).
Short Answer
Expert verified
Flux is 8 by both direct computation and the Divergence Theorem.
Step by step solution
01
Understand the Problem
We need to find the flux of the vector field \( \vec{F} = x^{2} \vec{i} + 2 y^{2} \vec{j} + 3 z^{2} \vec{k} \) across the closed surface \( S \) of a box with specified boundaries. The two methods to use are direct computation and the Divergence Theorem.
02
Define the Divergence Theorem
The Divergence Theorem states that for a vector field \( \vec{F} \) and a closed surface \( S \), \( \int_{S} \vec{F} \cdot d \vec{A} = \int_{V} abla \cdot \vec{F} \, dV \), where \( abla \cdot \vec{F} \) is the divergence of \( \vec{F} \) and \( V \) is the volume enclosed by \( S \).
03
Compute the Divergence of \( \vec{F} \)
The divergence of the vector field \( \vec{F} \) is \( abla \cdot \vec{F} = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial y}(2y^2) + \frac{\partial}{\partial z}(3z^2) \). Calculating the partial derivatives, we get: \( abla \cdot \vec{F} = 2x + 4y + 6z \).
04
Integrate the Divergence over the Volume
The volume \( V \) is given by the bounds \( 1 \leq x \leq 2 \), \( 0 \leq y \leq 1 \), \( 0 \leq z \leq 1 \). Compute \( \int_{1}^{2} \int_{0}^{1} \int_{0}^{1} (2x + 4y + 6z) \, dz \, dy \, dx \).
05
Evaluate the Integral
Calculate the inner integral: \( \int_{0}^{1} (6z) dz = [3z^2]_{0}^{1} = 3 \). The middle integral is \( \int_{0}^{1} (4y + 3) dy = [2y^2 + 3y]_{0}^{1} = 5 \). Finally, the outer integral is \( \int_{1}^{2} (2x + 5) dx = [x^2 + 5x]_{1}^{2} = (4 + 10) - (1 + 5) = 8 \). Thus, \( \int_{V} abla \cdot \vec{F} \, dV = 8 \).
06
Direct Flux Calculation on Each Face
Calculate the flux across each face of the box individually using \( \int \vec{F} \cdot d \vec{A} \). This approach involves summing all contributions from each face: two surfaces of each \( x \), \( y \), and \( z \) boundaries.
07
Calculate Flux for Each Face and Sum-Up
For face at \( x = 2 \), \( \vec{F} \approx 4 \vec{i} \), flux = \( 4 \). Face at \( x = 1 \), \( \vec{F} \) contributes \( -1 \). Similarly calculate for \( y = 1, y = 0, z = 1, z = 0 \) yielding contributions of 2, 0, 3, 0 respectively. Summing gives total flux = 8, consistent with the Divergence Theorem result.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Divergence Theorem
The Divergence Theorem is a powerful tool in vector calculus that connects the flow of a vector field through a closed surface to the behavior inside the surface. It provides a way to convert a surface integral into a volume integral. Imagining water flowing, the theorem essentially states that the total outward flow through a surface surrounding a volume equals the integral of the divergence over that volume. This helps simplify complex calculations because volume integrals are often easier to compute than surface integrals.
- Given a vector field \( \vec{F} \), the Divergence Theorem is stated as:\[ \int_{S} \vec{F} \cdot d\vec{A} = \int_{V} abla \cdot \vec{F} \, dV \]
- The left side represents the flux of the field through the surface \( S \), while the right side is the volume integral over the divergence of \( \vec{F} \).
Vector Field
A vector field assigns a vector to every point in a space. For example, \( \vec{F} = x^2 \vec{i} + 2y^2 \vec{j} + 3z^2 \vec{k} \) is a vector field that gives a unique vector at each \( (x, y, z) \) coordinate. Think of it as defining patterns of force or fluid flow in space. These fields are visualized by arrows pointing in the direction of the force or flow, with length proportional to its strength.
- Vector fields are crucial in physically representing concepts such as gravitational forces, velocity fields of fluids, or electromagnetic fields.
- They are often expressed as functions of the three spatial coordinates and can include terms for each of the x, y, and z directions.
Surface Integral
A surface integral allows us to compute quantities like mass, area, or flux across surfaces in three-dimensional spaces. For a vector field, the surface integral calculates the flow through a surface, integrating the dot product of the field vector and the differential area vector \( d\vec{A} \) over a surface.
- Expressed mathematically, for a vector field \( \vec{F} \), it's written as:\[ \int_{S} \vec{F} \cdot d\vec{A} \]
- This evaluates to the total quantity of interest (e.g., the total flux) passing through the surface \( S \).
- The direction of the surface's orientation is crucial; it often means deciding if the surface area vectors point outward or inward.
Partial Derivatives
Partial derivatives measure how a function changes as its input variables slightly change, holding all other variables constant. They are particularly useful in dealing with functions of several variables like those seen in vector fields.
- For example, the partial derivative of a vector field with respect to \( x \) might be written as:\[ \frac{\partial}{\partial x}(x^2) = 2x \]
- These derivatives help find the rate at which the components of the vector field change in each direction.
- These rates of change, when combined, give us the divergence of a field which informs us if the field is expanding or contracting at a point.
