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MA 242.003
• Day 51 – March 26, 2013• Section 13.1: (finish) Vector Fields• Section 13.2: Line Integrals
Chapter 13: Vector Calculus
Chapter 13: Vector Calculus
“In this chapter we study the calculus of vector fields,
…and line integrals of vector fields (work),
…and the theorems of Stokes and Gauss,
…and more”
Section 13.1: Vector Fields
Section 13.1: Vector Fields
Wind velocity vector field 2/20/2007
Section 13.1: Vector Fields
Wind velocity vector field 2/20/2007 Wind velocity vector field 2/21/2007
Section 13.1: Vector Fields
Ocean currents off Nova Scotia
Section 13.1: Vector Fields
Airflow over an inclined airfoil.
General form of a 2-dimensional vector field
General form of a 2-dimensional vector field
General form of a 2-dimensional vector field
Examples:
General form of a 2-dimensional vector field
Examples:
QUESTION: How can we visualize 2-dimensional vector fields?
General form of a 2-dimensional vector field
Examples:
Question: How can we visualize 2-dimensional vector fields?
Answer: Draw a few representative vectors.
Example:
Example:
Example:
We will turn over sketching vector fields in 3-space to MAPLE.
Gradient, or conservative, vector fields
Gradient, or conservative, vector fields
Gradient, or conservative, vector fields
EXAMPLES:
Gradient, or conservative, vector fields
EXAMPLES:
QUESTION: Why are conservative vector fields important?
QUESTION: Why are conservative vector fields important?
ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)
QUESTION: Why are conservative vector fields important?
ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)
Sections 13.2 and 13.3 are concerned with the following questions:
QUESTION: Why are conservative vector fields important?
ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)
Sections 13.2 and 13.3 are concerned with the following questions:
1. Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative.
QUESTION: Why are conservative vector fields important?
ANSWER: Because when combined with F = ma, leads to the law of conservation of total energy. ( in section 13.3)
Sections 13.2 and 13.3 are concerned with the following questions:
1. Given an arbitrary vector field, find a TEST to apply to the vector field to determine if it is conservative.
2. Once you know you have a conservative vector field, “Integrate it” to find its potential functions.
Format of chapter 13:
1. Sections 13.2, 13.3 - conservative vector fields
2. Sections 13.4 – 13.8 – general vector fields
Section 13.2: Line integrals
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.
EXAMPLE:
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
Line Integrals along piecewise differentiable curves
