Question

Define a line integral of a continuous vector field \(\mathbf{F}\) on a smooth curve \(C\). How do you evaluate the line integral as a definite integral?

Short answer

The line integral of a vector field \(\mathbf{F}\) over a smooth curve \(C\) can be evaluated as the definite integral \(\int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) dt\), where \(a\) and \(b\) are the limits of the parameter \(t\) that parameterizes the curve, \(\mathbf{r}(t)\) is the position vector and \(\mathbf{r}'(t)\) is its derivative with respect to \(t\).

Step-by-step solution

Understanding Line Integral

A line integral of a vector field \(\mathbf{F}\) along a smooth curve \(C\) in a plane or space is a way of accumulating the effect of the vector field along the curve. The line integral is often used in physics to compute work done by a force field for an object moving along a path, among other things.

Parametrizing the curve

Identify a parameterization for the path in the plane or space such that it provides a parameter (say, \(t\)) over which the curve is defined. Usually this parameterization is carried out in such a way that \(t\) represents time and \(\vec{r}(t)\) provides the position vector for each point \(t\) along the path of curve \(C\).

Decomposing the vector field

The given vector field \(\mathbf{F}\) needs to be decomposed into the fundamental unit vectors in the space (Generally \(i\), \(j\), \(k\) in 3D Space). These components are treated as functions of the parameters of the curve.

Evaluating the line Integral

The line integral is then calculated as a definite integral where the vector field is dotted with the derivative of the positional vector of the curve and integrated over the given interval of the parameter. This will be in the form: \(\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) dt\)