Unformatted text preview: r a “wedge” with the change in volume: Integrals in spherical coordinates: 44 15.10 Change of Variable in Multiple Integrals
Recall: Double Integral (Polar) We can view this as a change of variables Recall from Linear Algebra 2x2 and 3x3 matrices and their determinants
We can view the change of variables as transforming a complex shape in one coordinate system
to a simpler shape in another coordinate system
We assume the transformation is of class (functions with continuous partial derivatives) Definition (Jacobian)
Jacobian of the transformation Jacobian of the transformation 45 Theorem (Formula for Change of Variable in Double Integrals)
If :
1) The transformation is a class function 2) The Jacobian is nonzero
3) The transformation maps from one coordinate system to another
4) The function is continuous on the transformed to coordinate system
Then Theorem (Formula for Triple Integrals)
If :
1) The transformation is a class function 2) The Jacobian is nonzero
3) The transformation maps from one coordinate system to another
4) The function is continuous on the transformed to coordinate system
Then 46 Chapter 16: Vector Calculus 47 16.1 Vector Fields
In this chapter we are looking at vector functions that map from 2D to 2D, 3D to 3D and so
on and so forth
Definition (Vector Field):
Let is called a vector field in Let is called a vector field in Examples: Ocean current, wind map
Components:
2D:
3D:
The vector function is continuous if and only if the component functions are also continuous
Gradient Fields
2D: The gradient vector is orthogonal to the level curves at any point. This is called the gradient vector field
3D: The gradient vector is orthogonal to the level surfaces at any point. This is called the gradient vector field as well 48 Definition (Conservative Field)
A vector field is called a conservative field if there exists a scalar valued function such that: is called the potential function of 49 16.2 Line Integrals
Given:
1) A plane curve (
points) such that the function is continuous and nonzero at all 2) A function
We partition the curve into smaller pieces (partition the interval into smaller subintervals) and we
choose a point in the
interval: Definition (Line Integral)
We denote the line integral along the line with respect to the arclength function as: Recall: Arclength Generalization/Applications
1) Piecewise smooth curves
The line is the union of all the pieces 2) Space curves 50 3) Center of mass Line Integrals Along the Coordinate Axes: This is the line integral with respect to the arclength
With respect to x: With respect to y: Line Integrals for Vector Fields
Work in single variable calculus: Which is work done by a force f moving a particle from point a to b.
Now, what is the work done by the force
? in moving a particle along the curve 51 We can use line integrals to find the answer These are all equivalent expressions 52 16.3 Fundamental Theorem of Calculus for Line Integrals
Recall the fundamental theorem of calculus: Where the function is continuous over the interval
Theorem (Fundamental Theorem of Calculus for Line Integrals): Let...
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This note was uploaded on 02/03/2014 for the course CMPT 150 taught by Professor Dr.anthonydixon during the Spring '08 term at Simon Fraser.
 Spring '08
 Dr.AnthonyDixon

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