“Line” Integrals of Vector Fields

Vector Fields

So far we have discussed the following types of functions:

In Calc 1 (actualy in an Algebra class or a Pre-calculaus class) you learned about real functions of a single variable. These take one real number as input (single variable) and return a single real number (real function). Symbolically, this is written as \(f\colon \mathbb{R} \to \mathbb{R}\).

Earlier in Calc 3, we have learned about vector functions of a single variable: these take a single real number as input and return a vector as output. Symbolically we can write this as \(\mathbf{r}\colon \mathbb{R} \to \mathbb{R}^n\).

Also in Calc 3, we saw real functions of several variables. These take a point or vector as input, and return a single real number. Symbolically, \(f\colon\mathbb{R}^n \to \mathbb{R}\).

Now we are going to combine the last two ideas together and start looking at vector functions of several variables. For start, we will look at one simpler case where the dimension of the input is the same as the dimension of the output: functions from \(\mathbb{R}^n\) to \(\mathbb{R}^n\).

A function from \(\mathbb{R}^n\) to \(\mathbb{R}^n\) can be interpreted in different ways, some of which we will see later. For now, we will think about such functions as vector fields: the input is interpreted as a point in \(\mathbb{R}^n\), while the output is a vector. In other words, each point is the domain of the function has a vector associated with it.

Examples

Some examples of phenomena that can be modeled as vector fields:

Flow of a fluid: We saw that a vector function of one variable can be used to model a motion of a single object in a plane or space. In similar way we can model a flow of a liquid or gas on a plane or in a space: at each point, the flowing fluid has a velocity that determines the speed and direction of the flow. This velocity can be described by a vector. If the flow is constant in time, this will give us a vector field. If the flow changes with time, there will be an additional variable that represents time, and instead of a single vector field, we will get a parametrized family of vector fields. For each specific value of the time parameter, we would have a single vector field.
A force field: In the past we used vectors to model forces acting on objects. Vector fields can be used to model situations where there is a force at every point of a plane or space, like gravitational or electromagnetic force.
Gradient of a function: we have already seen one example of a vector field: the gradient vector of a function of several variables. At every point of the domain of \(f\), the vector \(\mathbf{\nabla} f\) points in the direction of the steepest increase of \(f\), and its magnitude gives us the rate of change of \(f\) in that direction.

Components of a vector field

A vector field \(\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3\) gives us a vector for each point \((x,y,z)\) in the domain of \(\mathbf{F}\). This vector can be written in components: \[\mathbf{F} = \left\langle P, Q, R\right\rangle = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}.\] The components \(P\), \(Q\) and \(R\) depend on the point \((x,y,z)\), and are therefore functions from \(\mathbb{R}^3\) to \(\mathbb{R}\), in other words, real functions of three variables. We can write it this way: \[\mathbf{F}(x,y,z) = \left\langle P(x,y,z), Q(x,y,z), R(x,y,z)\right\rangle = P(x,y,z)\mathbf{i} + Q(x,y,z)\mathbf{j} + R(x,y,z)\mathbf{k}\]

The functions \(P\), \(Q\) and \(R\) are called the component functions of the field \(\mathbf{F}\).

Differentiation of vector fields

Just like with vector functions, limits and therefore derivatives of vector fields are done by taking limits and derivatives of the component functions. Since vector fields and their component functions are functions of several variables, the derivatives will be directional derivatives and, in particular, partial derivatives. When all component function of a vector field are differentiable at a point, we say that the vector field is differentiable at the point.

Example:

Find the partial derivatives of the vector field \[\mathbf{F}(x,y) = \left\langle x^4y^3, x^2 + y^5\right\rangle\]

Here the component functions are \(P(x,y) = x^4y^3\) and \(Q(x,y) = x^2 + y^5\).

Partial derivative with respect to \(x\): \[\mathbf{F}_x(x,y) = \left\langle P_x(x,y), Q_x(x,y)\right\rangle = \left\langle 4x^3y^3, 2x\right\rangle\]
Partial derivative with respect to \(y\): \[\mathbf{F}_y(x,y) = \left\langle P_y(x,y), Q_y(x,y)\right\rangle = \left\langle 3x^4y^2, 5y^4\right\rangle\]

Later we will learn about some more interesting differential operators on vector fields.