VECTOR INTEGRAL CALCULUS
V1. Plane Vector Fields
Vector fields in the plane; gradient fields.
We consider a function of the type
where M and N are both functions of two variables. To each pair of values (xo, yo) for
which both M and N are defined, such a function assigns a vector F(xo, yo) in the plane.
F is therefore called a
vector function of two variables.
The set of points (x, y) for which
F is defined is called the domain of F.
To visualize the function F(x, y), at each point (xo, yo) in the domain we
place the corresponding vector F(xo, yo) so that its tail is at (xo, yo). Thus
each point of the domain is the tail end of a vector, and what we get is
This vector field gives a picture of the vector function
Conversely, given a vector field in a region of the xy-plane, it determines a vector function
of the type (I), by expressing each vector of the field in terms of its
Thus there is no real distinction between "vector function" and "vector field". Mindful of
the applications to physics, in these notes we will mostly use "vector field". We will use
the same symbol
F to denote both the field and the function, saying "the vector field F"
rather than "the vector field corresponding to the vector function F".
We say the vector field
F is continuous in some region of the plane if both M(x, y) and
N(x, y) are continuous functions in that region. The intuitive picture of a continuous vector
field is that the vectors associated to points sufficiently near (xo, yo) should have direction
and magnitude very close to that of F(xo, yo)
in other words, as you move around the
field, the vectors should change direction and magnitude smoothly, without sudden jumps
in size or direction.
In the same way, we say
F is differentiable in some region if M and N are differentiable,
that is, if all the partial derivatives
exist in the region. We say
F is continuously differentiable in the region if all these partial
derivatives are themselves continuous there. In general, all the commonly used vector fields
are continuously differentiable, except perhaps at isolated points, or along certain curves.
But as you will see, these points or curves affect the properties of the field in very important