Problem 12
Question
Find the curl and divergence of the given vector field. $$\left(x y^{2}, 3 y^{2} z^{2}, 2 x-z y^{3}\right)$$
Step-by-Step Solution
Verified Answer
The divergence of the given vector field is \( y^{2} + 6 y z^{2} - y^{3} \) and the curl of the given vector field is \( \left( 2y z^{2}, 2, 2y - 2xy \right) \).
1Step 1: Write down the vector field
The vector field is given by \( \mathbf{F} = x y^{2} \mathbf{i} + 3 y^{2} z^{2} \mathbf{j} + (2x - z y^{3}) \mathbf{k} \).
2Step 2: Compute the divergence of the vector field
The divergence of a vector field \( \mathbf{F} \) in Cartesian coordinates is given by \( \nabla \cdot \mathbf{F} = \frac{\partial F_{x}}{\partial x} + \frac{\partial F_{y}}{\partial y} + \frac{\partial F_{z}}{\partial z} \) . Applying this to our vector filed, we get \( \nabla \cdot \mathbf{F} = \frac{\partial (x y^{2})}{\partial x} + \frac{\partial (3 y^{2} z^{2})}{\partial y} + \frac{\partial (2x - z y^{3})}{\partial z} = y^{2} + 6 y z^{2} - y^{3} \).
3Step 3: Compute the curl of the vector field
The curl of a vector field \( \mathbf{F} = (F_{x}, F_{y}, F_{z}) \) in Cartesian coordinates is given by \( \nabla \times \mathbf{F} = \left(\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}, \frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x}, \frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}\right) \) . Applying this to our vector field, we obtain \( \nabla \times \mathbf{F} = \left( 2y z^{2} - 0, 2 - 0, 2y - 2xy \right) = \left( 2y z^{2}, 2, 2y - 2xy \right) \).
Key Concepts
Divergence of a Vector FieldCurl of a Vector FieldCartesian CoordinatesPartial Derivative
Divergence of a Vector Field
When exploring the concept of a vector field in calculus, one fundamental operation is to compute its divergence. The divergence provides a measure of how much a vector field spreads out from a given point. In mathematical terms, for a three-dimensional vector field represented as \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the divergence is calculated with the formula:
\[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]
In simpler terms, you take the partial derivatives of each component function relative to its respective variable and sum them up.
In our case, for the vector field \( \mathbf{F} = x y^{2} \mathbf{i} + 3 y^{2} z^{2} \mathbf{j} + (2x - z y^{3}) \mathbf{k} \), the divergence is calculated by taking the partial derivative of each component with respect to its corresponding variable—resulting in \( y^{2} + 6 y z^{2} - y^{3} \), which describes the rate at which the field is 'diverging' at each point in space.
\[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]
In simpler terms, you take the partial derivatives of each component function relative to its respective variable and sum them up.
In our case, for the vector field \( \mathbf{F} = x y^{2} \mathbf{i} + 3 y^{2} z^{2} \mathbf{j} + (2x - z y^{3}) \mathbf{k} \), the divergence is calculated by taking the partial derivative of each component with respect to its corresponding variable—resulting in \( y^{2} + 6 y z^{2} - y^{3} \), which describes the rate at which the field is 'diverging' at each point in space.
Curl of a Vector Field
The curl of a vector field is another important calculus concept, which helps to describe the rotation or 'twisting' behavior of the field around a point. For the same three-dimensional field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is found using the cross product of the del operator and the field itself:
\[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \]
Imagine a tiny paddle wheel placed in the flow of the vector field. The way this wheel tends to rotate gives you an idea of the field's curl. Applying this to the given field, \( \mathbf{F} = x y^{2} \mathbf{i} + 3 y^{2} z^{2} \mathbf{j} + (2x - z y^{3}) \mathbf{k} \), results in a curl of \( (2y z^{2}, 2, 2y - 2xy) \), indicating the rotation at various points within the field.
\[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \]
Imagine a tiny paddle wheel placed in the flow of the vector field. The way this wheel tends to rotate gives you an idea of the field's curl. Applying this to the given field, \( \mathbf{F} = x y^{2} \mathbf{i} + 3 y^{2} z^{2} \mathbf{j} + (2x - z y^{3}) \mathbf{k} \), results in a curl of \( (2y z^{2}, 2, 2y - 2xy) \), indicating the rotation at various points within the field.
Cartesian Coordinates
Cartesian coordinates form the backbone of vector field calculus. Invented by René Descartes, this coordinate system is defined by three axes—X, Y, and Z—orthogonal to one another, forming a three-dimensional space. Each point in this space can be uniquely identified by a set of three numbers \( (x, y, z) \) corresponding to its location along these axes.
In our vector field scenarios, we use Cartesian coordinates to express each point and to compute operations like divergence and curl. The spatial relationships dictated by these coordinates are crucial for determining how to apply partial derivatives to find the respective properties of the field.
In our vector field scenarios, we use Cartesian coordinates to express each point and to compute operations like divergence and curl. The spatial relationships dictated by these coordinates are crucial for determining how to apply partial derivatives to find the respective properties of the field.
Partial Derivative
Partial derivatives are essential to understanding vector fields. Unlike a regular derivative, which measures how a function changes as its input changes, a partial derivative does just that for multi-variable functions, but with respect to one variable at a time while holding the others constant. Formally, for a function \( f(x, y, z) \), its partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \).
This operation allows us to analyze the changes in the vector field in one direction independently of the others, which is central to calculating both the divergence and curl of the field. Understanding how each component of a vector field independently contributes to its overall behavior is a foundational aspect of vector field calculus.
This operation allows us to analyze the changes in the vector field in one direction independently of the others, which is central to calculating both the divergence and curl of the field. Understanding how each component of a vector field independently contributes to its overall behavior is a foundational aspect of vector field calculus.
Other exercises in this chapter
Problem 11
Evaluate the line integral. \(\int_{C} 3 y d x,\) where \(C\) is the half-ellipse \(x^{2}+4 y^{2}=4\) from (0,1) to (0,-1) with \(x \geq 0\)
View solution Problem 12
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
View solution Problem 12
Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). $$\begin{aligned} &\begin{array}{lllll} Q & \text { is } & \text {
View solution Problem 12
For a magnetic field \(\mathbf{B}\), Maxwell's equation \(\nabla \cdot \mathbf{B}=0\) implies that \(\mathbf{B}=\nabla \times \mathbf{A}\) for some vector field
View solution