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Suppose that the vector field
represents a force throughout a region
in the 3-dimensional space. Let
be a smooth curve in
The work done by this force over the curve
is equal to the integral
represents the (variable) radius vector from the origin to a point on
Assume that the curve
is given by the parametrization
is a direction vector of the tangent to
at the coresponding point.
be the curve with parametrization
Take a force (= a vector field)
- The force at a point on the curve:
- Scalar product:
- The work (=integration):