Multivariable calculus and vector analysis

Vector fields

Here’s a short review on vector fields.

A vector-valued function $\mathbf{F}:{\mathbf{R}}^2\to {\mathbf{R}}^2$ can be visualized as a vector field. At a point $\left(x,y\right)$, we plot the value of $\mathbf{F}\left(x,y\right)$ as a vector, such as in the following figure.

We repeat this over a set of points $\left(x,y\right)$ so that we can visualize the entire vector field.

For example, consider the function $\mathbf{F}\left(x,y\right)=\left(y,-x\right)$. We calculate values of the function at a set of points, such as

$$\begin{array}{llll}\hfill \mathbf{F}\left(1,0\right)& =\left(0,-1\right),& \hfill & \\ \hfill \mathbf{F}\left(0,1\right)& =\left(1,0\right),& \hfill & \\ \hfill \mathbf{F}\left(1,1\right)& =\left(1,-1\right),& \hfill & \\ \hfill \mathbf{F}\left(2,0\right)& =\left(0,-2\right),& \hfill & \end{array}$$

etc. By plotting these arrows, we see that this vector field appears to rotate in a clockwise direction.

(So that the vectors wouldn’t overlap, I drew the arrows at only 40% of the length they should have been. I scaled the length down so that the arrows would overlap; such scaling is typical in plots of vector fields.)

If we visualize the vector field $\mathbf{F}\left(x,y\right)=\left(x,y\right)$, it looks like an explosion emanating from the origin.

We can also plot vector fields in three dimensions, i.e., for functions $\mathbf{F}:{\mathbf{R}}^3\to {\mathbf{R}}^3$. For example, the three-dimensional analogue of the above picture would be $\mathbf{F}\left(x,y,z\right)=\left(x,y,z\right)$. Again, it corresponds to an explosion from the origin, although only a portion of this vector field with $x>0$, $y>0$, and $z>0$ is shown below.

We could get a rotation in three dimensions using $\mathbf{F}\left(x,y,z\right)=\left(y∕z,-x∕z,0\right)$. This is similar to the first vector field we plotted. In this case, since we divided by $z$, the magnitude of the vector field decreases as $z$ increases.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

4a. Vector field basics

• Vector fields
• The idea of divergence and curl
• The components of the curl
• More details on the components of the curl
• Divergence and curl notation
• Divergence and curl example
• Subtleties about divergence
• Subtleties about curl

4b. Parametrized curves

4c. Parametrized surfaces

4d. Path integrals and line integrals

4e. Surface integrals

5. The fundamental theorems of vector calculus

6. Review material