Differential calculus in higher dimension
WARNING
The information in this section is being updated.
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
Definition (inner product)
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)
For example, in
The primary higher-dimensional functions we consider in this course are:
- Scalar fields:
- Vector fields:
- Paths:
- Change of coordinates:
These possibilities all fit into the general pattern of
Open sets, closed sets, boundary, continuity
Let
Definition (interior point)
Let
Definition (open set)
A set
For example, open intervals, open disks, open balls, unions of open intervals, etc., are all open sets.
Lemma
Let
Proof
Let
Observe that the radius of the ball will be small for points close to the boundary.
Definition (Cartesian product)
If
Analogously the Cartesian product can be defined in higher dimensions: If
Lemma
If
Proof
Let
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
Observe that
Definition (boundary)
The set
Definition (closed)
A set
Lemma
Proof
Observe that
Limits and continuity
Let
Definition (Continuous)
A function
Even functions which look "nice" can fail to be continuous as we can see in the following example.
Example (continuity in higher dimensions)
Let
and
line | value |
---|---|
Theorem
Suppose that
, for every , , .
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
Since we already know that
Proof of part 4.
Take
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
Theorem
Let
Proof
We will independently prove the two implications.
- (
) Let , and observe that . We have already shown that the continuity of two vector fields implies the continuity of the inner product. - (
) By definition of the norm and we know as .
In higher dimensions the analogous statement is true for the vector field
Example (polynomials)
A polynomial in
E.g.,
Example (rational functions)
A rational function is a scalar field
where
As described in the following result, the continuity of functions continues to hold, in an intuitive way, under composition of functions.
Theorem
Suppose
makes sense. If
Proof
Example
We can consider the scalar field