Skip to content

Differential calculus in higher dimension

In this part of the course we work on the following skills:

  • Become comfortable working with coordinates in arbitrary dimension.
  • Develop an intuition for working with vector fields.
  • Understand the subtleties of derivatives in dimension greater than 1, evaluate and manipulate partial derivatives, directional derivatives, Jacobian.

See also the graded exercises and additional exercises associated to this part of the course.

Here we start to consider higher dimensional space. That is, instead of R we consider Rn for nN. We will particularly focus on 2D and 3D but everything also holds in any dimension. Going beyond R we have more options for functions and correspondingly more options for derivatives. Various different notation is commonly used. Here we will primarily use (x,y)R2, (x,y,z)R3 or, more generally, x=(x1,x2,,xn)Rn where x1R,,xnR. For example, R2 is the plane, R3 is 3D space.

Definition (inner product)

xy=k=1nxkykR

We recall that the inner product being zero has a geometric meaning, it means that the two vectors are orthogonal. We also recall that the "length" of a vector is given by the norm, defined as follows.

Definition (norm)

x=xx=(k=1nxk2)12.

For example, in R2 then (x,y)=x2+y2. There are various convenient properties for working with norms and inner products, in particular, the Cauchy-Schwarz inequality |xy|x y and the triangle inequality x+yx+y.

The primary higher-dimensional functions we consider in this course are:

  • Scalar fields: f:RnR
  • Vector fields: F:RnRn
  • Paths: α:RRn
  • Change of coordinates: x:RnRn

These possibilities all fit into the general pattern of f:RnRm for n,mN but tradition and use of the function gives us different terminology and symbols. Such functions are useful for representing various practical things, for example: gravitational force; temperature in a region; wind velocity; fluid flow; electric field; etc.

Open sets, closed sets, boundary, continuity

Let aRn, r>0. The open n-ball of radius r and centre a is written as

B(a,r):={xRn:xa<r}.

Definition (interior point)

Let SRn. A point aS is said to be an interior point if there is r>0 such that B(a,r)S. The set of all interior points of S is denoted intS.

Definition (open set)

A set SRn is said to be open if all of its points are interior points, i.e., if intS=S.

Interior points are the centre of a ball contained within the set

For example, open intervals, open disks, open balls, unions of open intervals, etc., are all open sets.

Lemma

Let r>0, aRn. The set B(a,r)Rn is open.

Proof

Let bB(a,r). It suffices to show that b is an interior point. (1) Let r1=ba<r. (2) Let r2=(rr1)/2. (3) We claim that B(b,r2)B(a,r): In order to see this take any cB(b,r2) and observe that

cacb+bar2+r1=r+r12<r.

Observe that the radius of the ball will be small for points close to the boundary.

Definition (Cartesian product)

If A1R, A2R then the Cartesian product is defined as

A1×A2:={(x,y):xA1,yA2}R2.

Analogously the Cartesian product can be defined in higher dimensions: If A1Rm, A2Rn then the Cartesian product A1×A2 is defined as the set of all points (x1,,xm,y1,,yn)Rm+n such that (x1,,xm)A1 and (y1,,yn)A2.

Lemma

If A1,A2 are open subsets of R then A1×A2 is an open subset of R2.

Proof

Let a=(a1,a2)A1×A2R2. Since A1 is open there exists r1>0 such that B(a1,r1)A1. Similarly for A2. Let r=min{r1,r2}. This all means that B(a,r)B(a1,r1)×B(a2,r2)A1×A2.

If A1,A2 are intervals then A1×A2 is a rectangle

Discussing the "interior" of the set naturally suggests the topic of the "boundary" of the set. In the following definitions we develop this idea.

Definition (exterior points)

Let SRn. A point aS is said to be an exterior point if there exists r>0 such that B(a,r)S=. The set of all exterior points of S is denoted extS.

Observe that extS is an open set. We use the notation Sc=RnS and we say that Cc is the complement of the set S.

Definition (boundary)

The set Rn(intSextS) is called the boundary of SRn and is denoted S.

Definition (closed)

A set SRn is said to be closed if SS.

Lemma

S is open Sc is closed.

Proof

Observe that Rn=intSSextS (disjointly). If xS then, for every r>0, B(x,r)S and so x(Sc). Similarly with S and Sc swapped and so S=(Sc). If S is open then intS=S and Sc=extSS=extS(Sc) and so Sc is closed. If S is not open then there exists aSS. Additionally a(Sc)S hence Sc is not closed.

Limits and continuity

Let SRn and f:SRm. If aRn, bRm we write limxaf(x)=b to mean that f(x)b0 as xa0. Observe how, if n=m=1, this is the familiar notion of continuity for functions on R.

Definition (Continuous)

A function f is said to be continuous at a if f is defined at a and limxaf(x)=f(a). We say f is continuous on S if f is continuous at each point of S.

Even functions which look "nice" can fail to be continuous as we can see in the following example.

Example (continuity in higher dimensions)

Let f be defined, for (x,y)(0,0), as

f(x,y)=xyx2+y2

and f(0,0)=0. What is the behaviour of f when approaching (0,0) along the following lines?

linevalue
{x=0}f(0,t)=0
{y=0}f(t,0)=0
{x=y}f(t,t)=12
{x=y}f(t,t)=12

Theorem

Suppose that limxaf(x)=b and limxag(x)=c. Then

  1. limxa(f(x)+g(x))=b+c,
  2. limxaλf(x)=λb for every λR,
  3. limxaf(x)g(x)=bc,
  4. limxaf(x)=b.

We prove a couple of the parts of the above theorem here, the other parts are left as exercises.

Proof of part 3.

Observe that f(x)g(x)bc=(f(x)b)(g(x)c)+b(g(x)c)+c(f(x)b). By the triangle inequality and Cauchy-Schwarz,

f(x)g(x)bcf(x)bg(x)c+bg(x)c+cf(x)b.

Since we already know that f(x)b0 and g(x)c0 as xa, this implies that f(x)g(x)bc0.

Proof of part 4.

Take f=g in part (c) implies that limxaf(x)2=b2.

When writing a vector field (or similar functions) it is often convenient to divide the higher-dimensional function into smaller parts. We call these parts the components of a vector field. For example F(x)=(F1(x),F2(x)) in 2D, F(x)=(F1(x),F2(x),F3(x)) in 3D, etc.

Theorem

Let F(x)=(F1(x),F2(x)). Then F is continuous if and only if F1 and F2 are continuous.

Proof

We will independently prove the two implications.

  • () Let e1=(1,0), e2=(0,1) and observe that Fk(x)=F(x)ek. We have already shown that the continuity of two vector fields implies the continuity of the inner product.
  • () By definition of the normF(x)F(a)2=k=12(Fk(x)Fk(a))2and we know Fk(x)Fk(a)0 as xa0.

In higher dimensions the analogous statement is true for the vector field F(x)=(F1(x),,Fm(x)) with exactly the same proof. I.e., F is continuous if and only if each fk is continuous.

Example (polynomials)

A polynomial in n variables is a scalar field on Rn of the form

f(x1,,xn)=k1=0jkn=0jck1,,knx1k1xnkn.

E.g., f(x,y):=x+2xyx2 is a polynomial in 2 variables. Polynomials are continuous everywhere in Rn. This is because they are the finite sum of products of continuous scalar fields.

Example (rational functions)

A rational function is a scalar field

f(x)=p(x)q(x)

where p(x) and q(x) are polynomials. A rational function is continuous at every point x such that q(x)0.

As described in the following result, the continuity of functions continues to hold, in an intuitive way, under composition of functions.

Theorem

Suppose SRl, TRm, f:SRm, g:TRn and that f(S)T so that

(gf)(x)=g(f(x))

makes sense. If f is continuous at aS and g is continuous at f(a) then gf is continuous at a.

Proof

limxaf(g(x))f(g(a))=limyg(a)f(y)f(g(a))=0

Example

We can consider the scalar field