0Introduction
IA Vector Calculus
0 Introduction
In the differential equations class, we learnt how to do calculus in one dimension.
However, (apparently) the world has more than one dimension. We live in a
3 (or 4) dimensional world, and string theorists think that the world has more
than 10 dimensions. It is thus important to know how to do calculus in many
dimensions.
For example, the position of a particle in a three dimensional world can be
given by a position vector
x
. Then by definition, the velocity is given by
d
dt
x
=
˙
x
.
This would require us to take the derivative of a vector.
This is not too difficult. We can just differentiate the vector componentwise.
However, we can reverse the problem and get a more complicated one. We can
assign a number to each point in (3D) space, and ask how this number changes
as we move in space. For example, the function might tell us the temperature at
each point in space, and we want to know how the temperature changes with
position.
In the most general case, we will assign a vector to each point in space. For
example, the electric field vector E(x) tells us the direction of the electric field
at each point in space.
On the other side of the story, we also want to do integration in multiple
dimensions. Apart from the obvious “integrating a vector”, we might want to
integrate over surfaces. For example, we can let
v
(
x
) be the velocity of some
fluid at each point in space. Then to find the total fluid flow through a surface,
we integrate v over the surface.
In this course, we are mostly going to learn about doing calculus in many
dimensions. In the last few lectures, we are going to learn about Cartesian
tensors, which is a generalization of vectors.
Note that throughout the course (and lecture notes), summation convention
is implied unless otherwise stated.