Multivariable calculus and vector analysis

Function notation

We have already used the notation that $\mathbf{R}$ stands for the real numbers. (Especially on the blackboard, we often use the notation $\mathbb{ℝ}$ .) Similarly, ${\mathbf{R}}^2$ is a two-dimensional vector, and ${\mathbf{R}}^3$ is a three-dimensional vector.

Scalar-valued functions

In one-variable calculus, you worked a lot with one-variable functions, i.e., functions from $\mathbf{R}$ onto $\mathbf{R}$. If $f\left(x\right)$ is such a one-variable functions, we can write $f:\mathbf{R}\to \mathbf{R}$ as a shorthand way of expressing that $f$ is a function from $\mathbf{R}$ onto $\mathbf{R}$.

A function like $f\left(x,y\right)=x+y$ is a function of two variables. It takes an element of ${\mathbf{R}}^2$, like $\left(2,1\right)$, and gives a value that is a real number (i.e., an element of $\mathbf{R}$), like $f\left(2,1\right)=3$. Since $f$ maps ${\mathbf{R}}^2$ to $\mathbf{R}$, we write $f:{\mathbf{R}}^2\to \mathbf{R}$. We can also use this “mapping” notation to define the actual function. We could define the above $f\left(x,y\right)$ by writing $f:\left(x,y\right)\mapsto x+y$.

To contrast a simple real number with a vector, we refer to the real number as a scalar. Hence, we can refer to $f:{\mathbf{R}}^2\to \mathbf{R}$ as a scalar-valued function of two variables or even just say it is a real-valued function of two variables.

Everything works the same for scalar valued functions of three or more variables. For example, $f\left(x,y,z\right)$, which we can write $f:{\mathbf{R}}^3\to \mathbf{R}$, is a scalar-valued function of three variables.

Vector-valued functions

In contrast, a vector-valued function takes on values that are vectors. First, let’s talk about vector-valued functions of a single variable.

A vector-valued function in two dimensions can be written $f:\mathbf{R}\to {\mathbf{R}}^2$. An example is $\mathbf{f}\left(t\right)=\left(3t,-t\right)$. For a given real number, which we’ll denote by $♣$ for fun, $\mathbf{f}\left(♣\right)$ is the two-dimensional vector $\left(3♣,-♣\right)$. Similarly, a vector-valued function in three dimensions can be written $f:\mathbf{R}\to {\mathbf{R}}^3$. For example, if $\mathbf{f}\left(s\right)=\left(1-s,s^3,coss\right)$, then $\mathbf{f}\left(0\right)=\left(1,0,1\right)$. We sometimes write vector-valued functions using the standard unit vector $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, as in $\mathbf{f}\left(s\right)=\left(1-s\right)\mathbf{i}+s^3\mathbf{j}+\left(coss\right)\mathbf{k}$.

Lastly, we can have vector-valued functions of multiple variables. For example, a function could take values in ${\mathbf{R}}^3$, say $\left(x,y,z\right)$, and map them to ${\mathbf{R}}^2$, such as $f\left(x,y,z\right)=\left(x-y,x^{22}∕z\right)$. We can write a function from ${\mathbf{R}}^3$ to ${\mathbf{R}}^2$ as $f:{\mathbf{R}}^3\to {\mathbf{R}}^2$. You get the idea.

The domain of a function

The function $f\left(t\right)=\left(t,t^2\right)$ is defined over all real numbers $\mathbf{R}$, i.e., the domain of the function is $\mathbf{R}$. Sometimes a function of one variable may be defined over a subset of real numbers, say some set $U\subset \mathbf{R}$; in this case, the domain of the function is $U$. (Note, the symbol “$\subset$” just means “is subset of”.) In three dimensions, for example, we can specify the domain by writing $\mathbf{f}:U\subset \mathbf{R}\to {\mathbf{R}}^3$, or simply $\mathbf{f}:U\to {\mathbf{R}}^3$.

Example: since $logt$ isn’t a real number for $t\le 0$, the domain of $\mathbf{f}\left(t\right)=\left(logt\right)\mathbf{i}+t\mathbf{j}$, is the set $D=\left(0,\infty \right)$. We could write this $\mathbf{f}:\left(0,\infty \right)\to {\mathbf{R}}^2$. What would the domain be if we replaced $logt$ with $log\left(t-3\right)$ or $log\left(2-t\right)$? You have to think where $log\left(t-3\right)$ or $log\left(2-t\right)$ is a real number, i.e., where $t-3>0$ or where $2-t>0$.

We use the same notation for functions of multiple variables. If we wrote $\mathbf{f}:U\subset {\mathbf{R}}^2\to {\mathbf{R}}^3$, we would mean a function maps values in a subset $U$ of ${\mathbf{R}}^2$ to values in ${\mathbf{R}}^3$,

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

• The cross product
• The component formula for the cross product
• The scalar triple product
• Cross product examples
• Scalar triple product example
• Spherical coordinates
• Examples of n-dimensional vectors
• Function notation

1b. Lines and planes

1c. Matrices

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material