ma 242.003 day 52 – april 1, 2013 section 13.2: line integrals – review line integrals of...
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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.
