Unit 1: Vector-Valued Functions
A vector-valued function (función de valor vectorial) is a function that returns vectors instead of points. We graph vector-valued functions by joining the extreme ends of all the vectors it returns.
Continuity can be defined for vector-valued functions, and they can also have:
Given any curve, we can calculate its arc length by using the derivative of its function and then integrating. We can parametrise a curve such that it is drawn at a speed that corresponds to its arc length; we call such a parametrisation an arc length parametrisation.
Vector-valued functions are especially useful in physics, since they help us describe an object’s position, and through their derivatives, its speed and acceleration, among many other physical concepts.
For instance, they help us describe the tangential and normal components of acceleration, which can, in turn, help us understand better certain phenomena, like the behaviour of a car on a curve.
Unit 2: Vector Fields
A vector field is a function that goes from to (a vector space of dimension ). The graphical representation of a vector fields involves drawing the vectors that the function returns, starting on the coordinates that were given to it.
A potential function of a vector field is a vector-valued function whose gradient vector is equal to said vector field. A conservative vector field is one that has (one or several) potential functions.
We can obtain the curl and divergence of a given vector field. These two operators help us know the behaviour of the vectors that the vector field returns:
- Curl tells us the rate of rotation of the vectors in a vector field.
- Divergence tells us how much the vectors “grow” or “shrink”
Additionally, the Laplacian of a vector-valued function is defined as the divergence of the function’s gradient vector.
Unit 3: Path and Line Integrals
The path integral (of a scalar function over a curve) is a special type of integral that has many applications, such as calculating the area of complex surfaces, calculating the average of a function over a curve, or calculating the centre of mass of an object.
The line integral (of a vector field over a curve) is another special type of integral. In general, it’s the equivalent of a path integral, but for vector fields instead of scalar functions. Line integrals also have many applications, such as calculating the work done by moving an object over a curve with a force defined by a vector field.
In all my notes, path integral refers to the integral of a scalar function over a curve, while line integrals refers to the integral of a vector field over a curve.
Some authors call both line integrals (of a scalar field and of a vector field, respectively).
Line integrals help us understand conservative vector fields (vector fields that are the gradient of some scalar function) better. The line integral of any conservative vector field over a curve will only depend on the two endpoints of the curve, with no regard for the actual chosen curve. We call this path independence.