Question

For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. $$\mathbf{v}=\left\langle 1-z^{2}, 0,0\right\rangle$$

Short answer

Based on the given vector field \(\mathbf{v} = \langle 1-z^2, 0, 0\rangle\), calculate the curl, make a sketch of the curl, and provide an interpretation of it.

Step-by-step solution

Recall the definition of curl

The curl of a vector field \(\mathbf{F} = \langle P, Q, R \rangle\) is defined as the cross product of the del operator \(\nabla\) with the vector field: $$ \nabla \times \mathbf{F} = \begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} $$

Compute the curl

Now, we apply the definition of curl to the given vector field \(\mathbf{v} = \langle 1-z^2, 0, 0\rangle\). Using the determinant of the matrix from the definition of curl, we get: $$ \nabla \times \mathbf{v} = \begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 1-z^2 & 0 & 0 \end{vmatrix} $$ Expanding the determinant, we get: $$ \nabla \times \mathbf{v} = \left\langle -\frac{\partial 0}{\partial y} + \frac{\partial 0}{\partial z}, -\frac{\partial 0}{\partial z} + \frac{\partial (1-z^2)}{\partial x}, -\frac{\partial 0}{\partial x} + \frac{\partial (1-z^2)}{\partial y} \right\rangle $$ After calculating the partial derivatives, the curl of the given vector field is: $$ \nabla \times \mathbf{v} = \left\langle 0,0,-\frac{\partial (1-z^2)}{\partial y} \right\rangle = \langle 0,0,0 \rangle $$

Sketch and interpret the curl

We found that the curl of the given vector field is \(\nabla \times \mathbf{v} = \langle 0,0,0 \rangle\). Since the curl is the zero vector, there is no rotation or circulation in the vector field. This can be seen graphically by sketching the vector field and noticing that there are no arrows pointing in the positive or negative z-direction, implying that there is no rotation or circulation present. The interpretation of the curl in this case is that the vector field has no rotational or circulatory motion.

Key concepts

Multivariable Calculus

Multivariable calculus is an extension of the single-variable calculus. It involves multiple variables and helps understand the relationships between them. In the context of vector fields, it allows us to handle functions of several variables. This branch of calculus is crucial for analyzing systems that depend on more than one input, like weather predictions or electromagnetic fields.
It introduces various operations, such as gradients, divergences, and curls, which are used to determine how a field behaves in three-dimensional space.
Using partial derivatives is key in multivariable calculus, which allows measuring rates of change in a function with respect to each variable independently. These derivatives are essential when computing the curl in vector fields, as they allow us to consider the different components individually.

Vector Calculus

Vector calculus is a field of mathematics that deals with vector fields and scalar fields. It extends concepts from calculus to vectors, which are quantities defined by both a magnitude and a direction. This is important in physics and engineering, where directions of forces and fields must be considered.
In vector calculus, operations like the curl, divergence, and gradient are crucial. Let's focus on curl, as in our exercise. Curl measures the amount of rotation or twisting about a point in a vector field. For a vector field \(\mathbf{F} = \langle P, Q, R \rangle\), the curl is expressed as \,\(abla \times \mathbf{F}\).

• The curl is a vector that describes the axis of rotation and how fast the field rotates around that axis.
• In physical terms, think of curling as swirling currents or rotating masses.

In the exercise, we computed the curl of a vector field, resulting in \(\langle 0,0,0 \rangle\), which indicates no rotation.

Vector Fields

A vector field assigns a vector to every point in a space. You can think of it like a map of arrows showing force and direction at every point in space. Mathematical weather maps and electromagnetic fields are examples of vector fields.
In the exercise, vector fields are described in three-dimensional space. The given vector field is \(\mathbf{v} = \langle 1-z^2, 0, 0\rangle\). This function assigns to each point \( (x, y, z) \) a vector directed along the x-axis with magnitude \(1-z^2\).

• This implies different regions exert different strengths of influence depending on the z value.
• The absence of y and z components means the field is uniform in these directions.

Understanding vector fields aids in visualizing how vectors influence the entire space, highlighting potential areas of high circulation or significant influence, though in this exercise there was none.