Problem 17
Question
Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. \(\operatorname{div} r=3\)
Step-by-Step Solution
Verified Answer
\( \operatorname{div} \mathbf{r} = 3 \), verified using the divergence formula.
1Step 1: Understand the Problem
We are given a vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) and need to verify that the divergence, \( \text{div} \, \mathbf{r} \), equals 3.
2Step 2: Recall the Divergence Formula
The divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is calculated as \( \text{div} \, \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).
3Step 3: Apply the Divergence Formula
For \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), we have \( P = x \), \( Q = y \), \( R = z \). Thus, \( \text{div} \, \mathbf{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} \).
4Step 4: Compute Each Partial Derivative
The partial derivatives are as follows: \( \frac{\partial x}{\partial x} = 1 \), \( \frac{\partial y}{\partial y} = 1 \), and \( \frac{\partial z}{\partial z} = 1 \).
5Step 5: Sum the Partial Derivatives
Add the partial derivatives: \( \text{div} \, \mathbf{r} = 1 + 1 + 1 = 3 \).
6Step 6: Verify the Conclusion
The computation shows that \( \text{div} \, \mathbf{r} = 3 \), which verifies the given identity.
Key Concepts
DivergencePartial DerivativesVector Fields
Divergence
Divergence is a measure of how much a vector field is "spreading out" or "converging" at a given point. Imagine a vector field as a collection of tiny arrows on a surface that indicate a direction and magnitude at every point. The divergence concept helps us to understand if these arrows are moving away from a point or towards it.
In mathematical terms, divergence is defined for a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \). It is calculated by taking the sum of the partial derivatives of each component of the vector field:
\[\text{div} \, \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]
This result tells us about the nature of the vector field in a region, such as if there is a source or sink at that point.
In mathematical terms, divergence is defined for a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \). It is calculated by taking the sum of the partial derivatives of each component of the vector field:
- \( \frac{\partial P}{\partial x} \): how \( P \), the \( x \)-component, changes as \( x \) changes.
- \( \frac{\partial Q}{\partial y} \): how \( Q \), the \( y \)-component, changes as \( y \) changes.
- \( \frac{\partial R}{\partial z} \): how \( R \), the \( z \)-component, changes as \( z \) changes.
\[\text{div} \, \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]
This result tells us about the nature of the vector field in a region, such as if there is a source or sink at that point.
Partial Derivatives
Partial derivatives allow us to understand how a multivariable function changes as one variable changes at a time, while the other variables are held constant. This is particularly useful in vector calculus, where we deal with functions of multiple variables.
Imagine you're on a hill. If you want to know how the steepness changes as you walk east (with your north/south position unchanged), you're looking at a partial derivative with respect to your east-west position.
Imagine you're on a hill. If you want to know how the steepness changes as you walk east (with your north/south position unchanged), you're looking at a partial derivative with respect to your east-west position.
- In our exercise, to find how \( P = x \) changes with respect to \( x \), we calculate \( \frac{\partial x}{\partial x} \), which is 1.
- Similarly, we compute \( \frac{\partial y}{\partial y} \) for \( Q = y \), and \( \frac{\partial z}{\partial z} \) for \( R = z \), both of which are also 1.
Vector Fields
A vector field assigns a vector to every point in a space, representing something like velocity, force, or any other vector quantity that varies within certain coordinates. This concept helps us visualize how a vector quantity acts in an entire region like a gravitational field or fluid flow.
Think of a vector field as a map for the forces within a region. Each point on the map has a vector arrow that shows the direction and strength of the force.
Think of a vector field as a map for the forces within a region. Each point on the map has a vector arrow that shows the direction and strength of the force.
- In our problem, the vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) represents a simple 3D vector field, where each component points along the \( x \), \( y \), and \( z \) axes.
- It helps to imagine each point in this space having a direction represented by this vector which changes with \( x, y, \) and \( z \).
Other exercises in this chapter
Problem 17
Assume that \(S\) forms the boundary of a closed and bounded region \(D\). If a is a constant vector, show that \(\iint_{S}(\mathbf{a} \cdot \mathbf{n}) d S=0\)
View solution Problem 17
Use Stokes' theorem to evaluate $$ \oint_{C} z^{2} e^{x^{2}} d x+x y^{2} d y+\tan ^{-1} y d z $$ where \(C\) is the circle \(x^{2}+y^{2}=9\), by finding a surfa
View solution Problem 17
Find the directional derivative of the given function at the given point in the indicated direction. $$ F(x, y, z)=x^{2} y^{2}(2 z+1)^{2} ;(1,-1,1),\langle 0,3,
View solution Problem 17
Find the curvature of an elliptical helix that is described by \(\mathbf{r}(t)=a \cos t \mathbf{i}+b \sin t \mathbf{j}+c t \mathbf{k}, a>0, b>0, c>0 .\)
View solution