Vector-Valued and Multivariable Functions

Part 3, Chapter 3: Calculus of Several Variables

Learning objectives

Define functions $f:\mathbb{R}^n\to\mathbb{R}^m$ via coordinate functions
Recognize parametric curves ( $n=1$ ) and parametric surfaces ( $n=2,m=3$ )
Distinguish vector fields ( $n=m$ ), scalar fields ( $m=1$ ), and curves ( $n=1$ )
Predict that smooth vector-valued functions are the natural objects of multivariable calculus

Once we leave the real line, almost every interesting function has more than one input or more than one output. A satellite's position is a triple $(x(t), y(t), z(t))$ as a function of one variable (time). A heat distribution is a single number $T(x, y, z)$ as a function of three. A robot arm's end-effector is a vector function of joint angles. The whole machinery of multivariable calculus, gradients, Jacobians, divergence, curl, rests on a single object: the vector-valued function. Master its bookkeeping in one variable and the higher-dimensional theory follows almost mechanically.

The definition in one line

A vector-valued function $f:\mathbb{R}^n\to\mathbb{R}^m$ assigns to each input $\mathbf{x}=(x_1,\ldots,x_n)$ an output vector $f(\mathbf{x})=(f_1(\mathbf{x}),\ldots,f_m(\mathbf{x}))$ mathbfx)). Each $f_i:\mathbb{R}^n\to\mathbb{R}$ i:mathbbRntomathbbR is a coordinate function. Everything about $f$ , continuity, differentiability, integrability, reduces to checking the same property for every $f_i$ i component-wise. That single principle is the most useful fact in the chapter.

Three special cases worth memorizing

Parametric curve ( $n=1$ ): $\mathbf{r}(t)=(x(t), y(t), z(t))$ traces a path through $\mathbb{R}^m$ . The tangent vector $\mathbf{r}'(t)$ is the instantaneous velocity.
Parametric surface ( $n=2,\ m=3$ ): $\mathbf{r}(u,v)=(x(u,v), y(u,v), z(u,v))$ sweeps out a 2D surface in 3D space.
Transformation ( $n=m$ ): The setting for the Inverse Function Theorem, a map from a space to itself. Coordinate changes (polar, spherical, cylindrical) are the canonical examples.

When $m=1$ we recover the scalar field $f:\mathbb{R}^n\to\mathbb{R}$ , whose graph is a surface (for $n=2$ ). The gradient $\nabla f$ , the Jacobian's special case, points uphill.

The grapher above lets you plot a single coordinate function of one variable. Try $\cos t$ and $\sin t$ side by side: each is a coordinate of the helix $\mathbf{r}(t)=(\cos t, \sin t, t)$ . The full helix lives in $\mathbb{R}^3$ but each of its components is a normal one-variable function. That is the entire trick.

Where this shows up

Robotics, forward kinematics: The end-effector position of a 6-axis arm is $\mathbf{p}(\boldsymbol{\theta})=(x,y,z,\text{roll},\text{pitch},\text{yaw})$ , a vector function of the six joint angles $\boldsymbol{\theta}\in\mathbb{R}^6$ . The inverse problem, find $\boldsymbol{\theta}$ for a desired $\mathbf{p}$ , is exactly the Inverse Function Theorem (§3.4) at work.
Computer-aided design (CAD): Every spline curve and Bézier surface in CAD is a parametric vector function. Solid modelling kernels evaluate millions of these per frame to render edges, intersections, and offsets.
Climate and fluid simulation: A wind field is $\mathbf{v}(x,y,z,t)=(u,v,w)$ , a 3-component vector function of 4 inputs (3 spatial + time). Numerical weather prediction integrates millions of grid points of this function at every timestep.

Pause and think: If $\mathbf{r}(t)=(\cos t, \sin t)$ traces the unit circle counterclockwise, what curve does $\mathbf{r}(t)=(\cos t, -\sin t)$ trace? Hint: compute $\mathbf{r}(0)$ and $\mathbf{r}(\pi/2)$ for each. Which direction does the second one go around?

Try it

Predict first: what shape is traced by $\mathbf{r}(t)=(2\cos t, 3\sin t)$ ? Verify by eliminating $t$ to get a Cartesian equation. (Hint: think about $(x/2)^2+(y/3)^2$ .)
Find the tangent vector to $\mathbf{r}(t)=(t,\ t^2,\ t^3)$ at $t=1$ . Sketch the direction in 3D. What does it tell you about the curve's direction of travel?
Predict: does the function $f(x,y)=(x+y,\ x-y)$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ cover all of $\mathbb{R}^2$ in its range? Justify by trying to solve $f(x,y)=(a,b)$ for arbitrary $a,b$ .
The parametric surface $\mathbf{r}(u,v)=(\cos u \sin v,\ \sin u \sin v,\ \cos v)$ for $u\in[0,2\pi),v\in[0,\pi]$ traces a familiar object. Which one? (Hint: compute $x^2+y^2+z^2$ .)

A trap to watch for

The graph of $f:\mathbb{R}^n\to\mathbb{R}^m$ lives in $\mathbb{R}^{n+m}$ , NOT in $\mathbb{R}^m$ . A function $\mathbb{R}^2\to\mathbb{R}^2$ has a graph that is a 2-dimensional surface in 4-dimensional space, impossible to draw directly. Beginners often try to "graph" such functions on a normal 2D plot, but the right picture is two separate scalar plots (one per coordinate function), or an arrow-field showing where each input vector gets sent. The mismatch between graph dimension and output dimension catches everyone the first time.

What you now know

You can write any vector-valued function in coordinate form and read off whether it is a curve, surface, scalar field, or transformation. You know the next four sections all extend the one-variable tools, limits, continuity, derivatives, and their inverses, component-by-component to this richer setting.

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References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 3.
Spivak, M. (1965). Calculus on Manifolds. W. A. Benjamin, ch. 2.
Munkres, J. R. (1991). Analysis on Manifolds. Westview Press, ch. 1.
Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill, ch. 9.
Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley, ch. 12.