Problem 6
Question
Find the curl and divergence of the given vector field. $$\left\langle x e^{z}, y z^{2}, x+y\right\rangle$$
Step-by-Step Solution
Verified Answer
The curl of the given vector field is \( curl(F) = 2z \mathbf{i} - (1 - z)\mathbf{j} - \mathbf{k} \) while the divergence of the vector field is \( div(F) = 2z+1\).
1Step 1: Definition of the Vector Field
The vector field F is defined as \( F = e^{z} \mathbf{i} + y z^{2} \mathbf{j} + (x+y) \mathbf{k} \).
2Step 2: Compute Curl of Vector Field
To compute the curl of vector field, use the formula: \[ curl(F) = (\nabla x F) = ((\frac{\partial}{\partial y}(x+y) - \frac{\partial}{\partial z}(yz^2))\mathbf{i} - (\frac{\partial}{\partial x}(x+y) - \frac{\partial}{\partial z}(e^z))\mathbf{j} + (\frac{\partial}{\partial x}(yz^2) - \frac{\partial}{\partial y}(e^z))\mathbf{k} \] On computing the required partial derivatives, we have: \[ curl(F) = (2z - 0)\mathbf{i} - (1 - z)\mathbf{j} + (0 - 1)\mathbf{k} = 2z \mathbf{i} - (1 - z)\mathbf{j} - \mathbf{k} \] Hence, the curl of the vector field \( F \) is \( curl(F) = 2z \mathbf{i} - (1 - z)\mathbf{j} - \mathbf{k} \).
3Step 3: Compute Divergence of Vector Field
Next, to compute the divergence of the vector field, we use the formula: \[ div(F) = \nabla . F = \frac{\partial}{\partial x}(e^z) + \frac{\partial}{\partial y}(yz^2) + \frac{\partial}{\partial z}(x+y) \] On computing the required partial derivatives, we have: \[ div(F) = 0 + 2z + 1 = 2z + 1 \] Hence, the divergence of the vector field \( F \) is \( div(F) = 2z + 1\).
Key Concepts
Curl of Vector FieldDivergence of Vector FieldPartial Derivatives
Curl of Vector Field
In vector calculus, the curl of a vector field is a measure of the rotation or the "twist" of the field at a point. It tells us how much and in what direction the field "curls" around that point. This is particularly useful in physics for understanding rotational flow patterns, such as those in a fluid or electromagnetic field.
To find the curl of a vector field \( \mathbf{F} \), we use the formula:
The curl of the vector field \( \langle x e^{z}, y z^{2}, x+y \rangle \) was calculated by taking the cross product of the del operator with the vector field. The result is:
\[\text{curl}(\mathbf{F}) = 2z \mathbf{i} - (1 - z)\mathbf{j} - \mathbf{k}\]
This shows that the field has a tendency to rotate about the axis defined by these components, which can vary based on the value of \( z \).
To find the curl of a vector field \( \mathbf{F} \), we use the formula:
- \( \text{curl}(\mathbf{F}) = abla \times \mathbf{F} \)
The curl of the vector field \( \langle x e^{z}, y z^{2}, x+y \rangle \) was calculated by taking the cross product of the del operator with the vector field. The result is:
\[\text{curl}(\mathbf{F}) = 2z \mathbf{i} - (1 - z)\mathbf{j} - \mathbf{k}\]
This shows that the field has a tendency to rotate about the axis defined by these components, which can vary based on the value of \( z \).
Divergence of Vector Field
The divergence of a vector field provides a scalar value that represents the magnitude of a source or sink at a given point in the field. Think of it as a measure of how much the field "spreads out" from a given point. In simpler terms, it quantifies the "density" of the flow exiting an infinitesimally small region surrounding the point.
To compute the divergence, you use the dot product with the del operator as follows:
\[\text{div}(\mathbf{F}) = 0 + 2z + 1 = 2z + 1\]
This suggests a net outflow or inflow pattern depending on the value of \( z \), contributing to a larger expansion or contraction at various regions of the space.
To compute the divergence, you use the dot product with the del operator as follows:
- \( \text{div}(\mathbf{F}) = abla \cdot \mathbf{F} \)
\[\text{div}(\mathbf{F}) = 0 + 2z + 1 = 2z + 1\]
This suggests a net outflow or inflow pattern depending on the value of \( z \), contributing to a larger expansion or contraction at various regions of the space.
Partial Derivatives
Partial derivatives are a fundamental tool in vector calculus, useful for analyzing functions with multiple variables. They allow us to understand how a function changes as we vary one of the input variables, keeping others constant.
For a scalar function \( f(x, y, z) \), the partial derivative with respect to \( x \) is denoted by \( \frac{\partial f}{\partial x} \). It represents the rate of change of the function as we slightly vary \( x \) while keeping \( y \) and \( z \) fixed.
In the context of vector fields, these derivatives are necessary for computing both curl and divergence.
For a scalar function \( f(x, y, z) \), the partial derivative with respect to \( x \) is denoted by \( \frac{\partial f}{\partial x} \). It represents the rate of change of the function as we slightly vary \( x \) while keeping \( y \) and \( z \) fixed.
In the context of vector fields, these derivatives are necessary for computing both curl and divergence.
- For curl, we need derivatives that "mix" the different components—such as \( \frac{\partial}{\partial y}(x+y) \) or \( \frac{\partial}{\partial z}(yz^2) \).
- For divergence, you sum the derivatives like \( \frac{\partial}{\partial x}(e^z) \), \( \frac{\partial}{\partial y}(yz^2) \), and \( \frac{\partial}{\partial z}(x+y) \).
Other exercises in this chapter
Problem 5
Evaluate the line integral. \(\int_{C} 2 x d x,\) where \(C\) is the line segment from (0,2) to (2,6)
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Use Stokes' Theorem to compute $$\begin{aligned}&\iint(\nabla \times \mathbf{F}) \cdot \mathbf{n} d \mathbf{S}\\\&S \end{aligned}$$ \(S\) is the portion of the
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Sketch several vectors in the vector field by hand and verify your sketch with a CAS. $$\mathbf{F}(x, y)=-\mathbf{i}+y^{2} \mathbf{j}$$
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Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). \(Q\) is bounded by \(4 x+2 y-z=4(z \leq 0)\) and the coordinate p
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