MA 242.003

• 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:

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:

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

Extension to 3-dimensional space

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?

What is the geometrical interpretation of the line integral?

What is the geometrical interpretation of the line integral?

(continuation of example)

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

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

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.

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.

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.

Change in notation for line integrals of vector fields.