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
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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.
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We partition the curve into n pieces:
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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:
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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:
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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.
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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.
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Extension to 3-dimensional space
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Extension to 3-dimensional space
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Extension to 3-dimensional space
Shorthand notation
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Extension to 3-dimensional space
Shorthand notation
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Extension to 3-dimensional space
Shorthand notation
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Extension to 3-dimensional space
Shorthand notation
3. Then
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What is the geometrical interpretation of the line integral?
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What is the geometrical interpretation of the line integral?
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What is the geometrical interpretation of the line integral?
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(continuation of example)
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A major application: Line integral of a vector field along C
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A major application: Line integral of a vector field along C
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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.
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Principle: Only the component of force in the direction of motion contributes to the motion.
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Principle: Only the component of force in the direction of motion contributes to the motion.
Direction of motion
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Principle: Only the component of force in the direction of motion contributes to the motion.
Direction of motion
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Principle: Only the component of force in the direction of motion contributes to the motion.
Direction of motion
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Partition C into n parts, and choose sample points in each sub – arc.
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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.
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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
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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.
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which is a Riemann sum!
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which is a Riemann sum! We define the work as the limit as .
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Change in notation for line integrals of vector fields.
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Change in notation for line integrals of vector fields.
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