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## Work of a force.

Definition 8.11   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

where represents the (variable) radius vector from the origin to a point on .

Assume that the curve is given by the parametrization , where . Then is a direction vector of the tangent to at the coresponding point.

Example 8.12   Let be the curve with parametrization . Take a force (= a vector field) .
• The force at a point on the curve:

• Tangent:

• Scalar product:

• The work (=integration):

Example 8.13

Noah Dana-Picard
2001-05-30