Vector-Valued and Multivariable Functions

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 $f(\\mathbf{x})=(f\_1(\\mathbf{x}),\\ldots,f\_m(\\mathbf{x}))$. Each $f\_i:\\mathbb{R}^n\\to\\mathbb{R}$ is a coordinate function. Everything about $f$, continuity, differentiability, integrability, reduces to checking the same property for every $f\_i$ 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.

• 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 (Section 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.

Mark section complete →

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.