Question

What is a conservative vector field and how do you test for it in the plane and in space?

Short answer

A conservative vector field is one where the line integral is path independent. To test whether a vector field is conservative, in both the plane and in space, compute the curl of the vector field. If the curl equals zero in the plane and is a zero vector in space, then the vector field is conservative.

Step-by-step solution

Definition of Conservative Vector Field

A Conservative vector field is a vector field where the line integral from one point to another point is independent of the path taken. Practically, this means that if you were to move in the field starting from one point and eventually return to that same point, the overall 'work' done or 'energy' needed would be zero. It's like going around a hill (which stands for the vector field); the work done up the slope will be recouped on the slope down, so the net energy/doccomplished work is zero.

Testing in the plane

In the plane, a vector field is conservative if and only if it has zero curl. The curl (which in the plane is a scalar quantity) of a vector field \(F = M(x, y) \hat{i} + N(x, y) \hat{j}\) is defined as \(\nabla \times F = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\). Hence, to test whether the vector field is conservative or not, find the curl of the vector field using the formula and if it equates to zero then the vector field is conservative.

Testing in the space

In the space, you would take a similar approach but with a slight difference. Here, the curl is a vector quantity. The curl of a vector field \(F = M(x, y, z) \hat{i} + N(x, y, z) \hat{j} + P(x, y, z) \hat{k}\) is defined as \(\nabla \times F = \left(\frac{\partial P}{\partial y} - \frac{\partial N}{\partial z}\right) \hat{i} + \left(\frac{\partial M}{\partial z} - \frac{\partial P}{\partial x}\right) \hat{j} + \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) \hat{k} \). So, you need to compute the curl using this formula and if the resultant vector is a zero vector, then the vector field is said to be conservative.