MA Day 52 – April 1, 2013 Section 13.2: Line Integrals – Review line integrals of f(x,y,z) – Line integrals of vector fields

Section 13.2: Line integrals GOAL: To generalize the Riemann Integral of f(x) along a line to an integral of f(x,y,z) along a curve in space.

We partition the curve into n pieces:

Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum:

which is similar to a Riemann sum.

Assuming f(x,y) is continuous, we evaluate it at sample points, multiply by the arc length of each subarc, and form the following sum: which is similar to a Riemann sum.

Extension to 3-dimensional space

Shorthand notation

Extension to 3-dimensional space Shorthand notation

Extension to 3-dimensional space Shorthand notation

Extension to 3-dimensional space Shorthand notation 3. Then

What is the geometrical interpretation of the line integral?

(continuation of example)

A major application: Line integral of a vector field along C

We generalize to a variable force acting on a particle following a curve C in 3-space.

Principle: Only the component of force in the direction of motion contributes to the motion.

Direction of motion

Principle: Only the component of force in the direction of motion contributes to the motion. Direction of motion

Principle: Only the component of force in the direction of motion contributes to the motion. Direction of motion

Partition C into n parts, and choose sample points in each sub – arc.

Notice that the unit tangent vector T gives the instantaneous direction of motion.

Partition C into n parts, and choose sample points in each sub – arc. Notice that the unit tangent vector T gives the instantaneous direction of motion. Remembering the work done formula

Partition C into n parts, and choose sample points in each sub – arc. Notice that the unit tangent vector T gives the instantaneous direction of motion.

which is a Riemann sum!

which is a Riemann sum! We define the work as the limit as.

Change in notation for line integrals of vector fields.