Line integrals in a vector field
What we are building to
- The vector from the field at the point where you are standing.
- The displacement vector associated with the next step you take along this curve.
Whale falling from the sky
Example 1: Putting numbers on Whilly's fall.
Visualizing more general line integrals through a vector field
- is a vector field, associating each point in space with a vector. You can think of this as a force field.
- is a curve through space.
- is a vector-valued function parameterizing the curve in the range
- is the derivative of , representing the velocity vector of a particle whose position is given by while increases at a constant rate. When you multiply this by a tiny step in time, , it gives a tiny displacement vector, which I like to think of as a tiny step along the curve. Technically it is a tiny step in the tangent direction to the curve, but for small enough this amounts to the same thing.
- Note, in this animation the length of stays constant. This is not necessarily true for most parameterizations of , which may have you speeding up or slowing down as your position varies according to . For example, Whilly was probably speeding up during his fall, making the velocity vector grow over time.
- The rotating circle in the bottom right of the diagram is a bit confusing at first. It represents the extent to which the vector lines up with the tangent vector . The grey and vectors are shown to see how these vectors are oriented relative to -plane as a whole.
Example 2: Work done by a tornado
Step 1: Expand the integral
Step 2: Expand each component
Step 3: Solve the integral
- The shorthand notation for a line integral through a vector field is
- Line integrals are useful in physics for computing the work done by a force on a moving object.
- If you parameterize the curve such that you move in the opposite direction as increases, the value of the line integral is multiplied by .