Question

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

Short answer

Based on the given velocity field, \(\mathbf{v} = \langle -2z, 0, 1 \rangle\), the curl of the velocity field was found to be \(\nabla \times \mathbf{v} = \left\langle 0, -2, 0 \right\rangle\). A sketch of the curl shows that it is a constant vector field with arrows parallel to the negative y-axis, equally spaced, and with the same magnitude. The curl of the velocity field indicates that there is a rotation around the negative y-direction (counterclockwise when looking in the negative y-direction) with a magnitude of 2 at every point in space. This means that the original velocity field has a uniform rotational motion around the negative y-axis with a constant magnitude.

Step-by-step solution

Compute the curl of the velocity field

The curl of a vector field \(\mathbf{v} = \langle P, Q, R \rangle\) is defined as the following cross product: $$\nabla \times \mathbf{v} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{array} \right|$$ Given the velocity field: \(\mathbf{v}=\langle -2z, 0, 1 \rangle\), let's compute its curl: $$\nabla \times \mathbf{v} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ -2z & 0 & 1 \end{array} \right| = \left\langle 0, \frac{\partial}{\partial x} (1) - \frac{\partial}{\partial z} (-2z), \frac{\partial}{\partial y} (-2z) - \frac{\partial}{\partial x} (0) \right\rangle$$ Now, let's find the partial derivatives: $$\frac{\partial}{\partial x} (1) = 0$$ $$\frac{\partial}{\partial z} (-2z) = -2$$ $$\frac{\partial}{\partial y} (-2z) = 0$$ So, the curl of the velocity field is: $$\nabla \times \mathbf{v} = \left\langle 0, -2, 0 \right\rangle$$

Sketch the curl

To make a sketch of the curl, we need to recognize that the curl is a constant vector field, pointing in the negative y-direction with a magnitude of 2. For an accurate sketch, we can use a 3D plot tool. However, roughly, the curl of the velocity field would look like many arrows parallel to the negative y-axis, equally spaced, all with the same magnitude.

Interpret the curl

The curl of a vector field, in general, represents the rotation and/or circulation of the vector field around a point. In this case, we have a curl that is a constant vector field with components \(\left\langle 0, -2, 0 \right\rangle\), which indicates that there is a rotation around the negative y-direction (counterclockwise when looking in the negative y-direction) at every point in space with a magnitude of 2. This means that the original velocity field \(\mathbf{v} = \langle -2z, 0, 1 \rangle\) has a uniform rotational motion around the negative y-axis with a constant magnitude.

Key concepts

Multivariable Calculus

Multivariable calculus, also known as multiple-integral calculus, extends the concepts of single-variable calculus to higher dimensions. This branch of mathematics is essential for understanding and describing phenomena in spaces that are more complex than the simple one-dimensional space we start with in introductory calculus.

For example, if one were to study how heat transfers across different points on a 3D object, variables such as temperature and spatial coordinates in three dimensions would come into play. Multivariable calculus provides the tools to describe rates of change and accumulation in such multi-dimensional contexts. Central concepts in this field involve vector fields, partial derivatives, and multiple integrals, which allow us to compute quantities like flux through a surface or the curl of a vector field, as in the given exercise.

Vector Calculus

In the realm of vector calculus, we extend basic calculus notions to vector fields, which are functions that assign a vector to every point in space. For those new to the concept, think of vector fields like a weather map indicating wind velocity at various locations—each point has a direction and magnitude represented by an arrow.

Visualizing Vector Fields

Visualizing these fields is crucial for conceptual understanding. One may use arrows to represent the vectors' directions and lengths to demonstrate their magnitudes in a 2D or 3D plot. When it comes to the curl of a vector field, we are looking at the field's tendency to induce rotation around a point or along a path, which is critical for understanding fluid flow, electromagnetic fields, and mechanics.

Partial Derivatives

Partial derivatives are one of the cornerstones of multivariable calculus. When you're dealing with functions of several variables, partial derivatives tell us how a function changes as we vary just one of those variables, while keeping the others fixed.

Consider a terrain, where the height of the land is a function of two horizontal variables: latitude and longitude. The partial derivative with respect to latitude tells you how steep the terrain is if you walk directly north or south, while the partial derivative with respect to longitude indicates the slope experienced when walking east or west. These concepts empower us to delve into more complex analyses of change, such as when calculating the curl of a vector field, which fundamentally involves computing partial derivatives as shown in our exercise solution.