Problem 1
Question
Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; \sigma \text { is the surface of the cube }} \\ {\text { bounded by the planes } x=0, x=1, y=0, y=1, z=0} \\ {z=1}\end{array} $$
Step-by-Step Solution
Verified Answer
Both integrals give 3, aligning with the Divergence Theorem.
1Step 1: Understand the Divergence Theorem
The Divergence Theorem states that the flux of a vector field \( \mathbf{F} \) across a closed surface \( \sigma \) is equal to the triple integral of the divergence of \( \mathbf{F} \) over the volume \( V \) enclosed by \( \sigma \). Mathematically, this is \( \iint_{\sigma} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} abla \cdot \mathbf{F} \, dV \).
2Step 2: Calculate the Divergence
Calculate the divergence \( abla \cdot \mathbf{F} \) of the vector field \( \mathbf{F}(x, y, z) = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \). We have: \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x} x + \frac{\partial}{\partial y} y + \frac{\partial}{\partial z} z = 1 + 1 + 1 = 3 \).
3Step 3: Evaluate the Triple Integral
Compute the triple integral over the volume of the cube for \( abla \cdot \mathbf{F} = 3 \). The limits for \( x, y, z \) are from 0 to 1. Thus, \( \iiint_{V} 3 \, dV = 3 \int_0^1 \int_0^1 \int_0^1 \, dx \, dy \, dz = 3 \cdot 1 = 3 \).
4Step 4: Evaluate the Surface Integral
Parametrize each face of the cube and calculate \( \mathbf{F} \cdot d\mathbf{S} \) for each one. For example, on the face \( x = 0 \), \( d\mathbf{S} = -\mathbf{i} \, dy \, dz \) and \( \mathbf{F} \cdot d\mathbf{S} = 0 \). Calculate this for all 6 faces. You will find: \( \int_{x=0} = 0, \int_{x=1} = 1, \int_{y=0} = 0, \int_{y=1} = 1, \int_{z=0} = 0, \int_{z=1} = 1 \). The total flux is \( 3 \).
5Step 5: Compare Results
Compare the results from the surface integral and triple integral. Both results are \( 3 \), verifying that the Divergence Theorem holds for the given vector field \( \mathbf{F} \) and the cube surface \( \sigma \).
Key Concepts
Surface Integrals in Vector CalculusUnderstanding Triple IntegralsExploring Vector Fields
Surface Integrals in Vector Calculus
In vector calculus, a surface integral allows us to understand how a vector field interacts with a surface. Consider the surface of a shape, like the cube from our original exercise. Here, the concept is about checking how the vector field passes through or along this surface. The vector field in question is \( \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} \).
A surface integral effectively computes the total of a field’s interaction with a surface, by summing up all the little pieces over the surface. The process involves taking the dot product of the vector field and a small piece of the surface area, \( d\mathbf{S} \). Integrating this over all small pieces on the surface gives the full result.
This is particularly important when validating the Divergence Theorem, which states that such an integral over a closed surface should match the triple integral over the volume inside. Each face of a closed shape, like a cube, has to be considered. So, for the cube, the surface integral involves separate calculations over each of its six faces.
A surface integral effectively computes the total of a field’s interaction with a surface, by summing up all the little pieces over the surface. The process involves taking the dot product of the vector field and a small piece of the surface area, \( d\mathbf{S} \). Integrating this over all small pieces on the surface gives the full result.
This is particularly important when validating the Divergence Theorem, which states that such an integral over a closed surface should match the triple integral over the volume inside. Each face of a closed shape, like a cube, has to be considered. So, for the cube, the surface integral involves separate calculations over each of its six faces.
Understanding Triple Integrals
Triple integrals go beyond ordinary or double integrals by adding another dimension. They are essential for computing quantities over three-dimensional spaces, as shown in our original problem where we computed over the cube's volume. When we computed \( \iiint_{V} 3 \, dV \), this meant we integrated over the space enclosed from 0 to 1 in the \( x \), \( y \), and \( z \) directions.
Triple integrals allow you to accumulate values across a 3D region. Here, it helped us sum the divergence over the entire volume of the cube. This integration respects the limits for each axis, and in simple scenarios, it involves straightforward multiplicative computation, like multiplying by dimensions of the integration region.
In context, this triple integral directly connects to the Divergence Theorem. It validates the precept that the sum of the divergence over the volume equals the flux across the surface. As seen, the integral simplification gave the volume's contribution to be 3, matching exactly with the surface integral.
Triple integrals allow you to accumulate values across a 3D region. Here, it helped us sum the divergence over the entire volume of the cube. This integration respects the limits for each axis, and in simple scenarios, it involves straightforward multiplicative computation, like multiplying by dimensions of the integration region.
In context, this triple integral directly connects to the Divergence Theorem. It validates the precept that the sum of the divergence over the volume equals the flux across the surface. As seen, the integral simplification gave the volume's contribution to be 3, matching exactly with the surface integral.
Exploring Vector Fields
A vector field assigns a vector to every point in space, visually resembling arrows pointing in various directions and magnitudes. The vector field here, \( \mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} \), assigns vectors pointing diagonally from the origin outward.
Vector fields are essential when applying theorems like the Divergence Theorem because they define how the field behaves across an area or through a volume. Each component of a vector field is typically associated with partial derivatives, crucial when calculating divergence, which measures how much the field is "spreading out."
Understanding vector fields is key for solving problems involving force fields, fluid flow, or electromagnetic fields. In our exercise, the diagonal nature of the vector field indicated symmetry that simplified computations. Recognizing how vector fields influence these integrations ensures smooth application of relevant mathematical theorems.
Vector fields are essential when applying theorems like the Divergence Theorem because they define how the field behaves across an area or through a volume. Each component of a vector field is typically associated with partial derivatives, crucial when calculating divergence, which measures how much the field is "spreading out."
Understanding vector fields is key for solving problems involving force fields, fluid flow, or electromagnetic fields. In our exercise, the diagonal nature of the vector field indicated symmetry that simplified computations. Recognizing how vector fields influence these integrations ensures smooth application of relevant mathematical theorems.
Other exercises in this chapter
Problem 1
Verify Formula (2) in Stokes’ Theorem by evaluating the line integral and the surface integral. Assume that the surface has an upward orientation.. $$ \begin{ar
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Let \(C\) be the line segment from \((0,0)\) to \((0,1) .\) In each part, evaluate the line integral along \(C\) by inspection, and explain your reasoning. $$ \
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