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Unformatted text preview: MA1104 Multivariable Calculus Lecture 15 Dr KU Cheng Yeaw Monday March 16, 2009 Recap Last Lecture ... we finished our study on multiple integrations! Overview In this lecture, • we begin a study on vector calculus. We first introduce vector fields. Unlike (scalar) functions of several variables, a vector field assigns a vector to a point in plane/space. • we introduce line integrals over a curve in plane/space as a generalization of single integral over an interval. Vector Field So far, we have developed the calculus of (scalar) functions of several variables. On many instances, we borrow intuition from Single Variable Calculus. We have also seen the role and usefulness of vector functions r ( t ) in the study of scalar functions. In this last chapter of MA1104, we shall study the calculus of vector fields. These are functions that assign vectors to points in plane or space. We shall define • Line integrals  integrals which can be used to find the work done by a force field in moving an object along a curve. • Surface integrals  integrals which can be used to find the rate of fluid flow across a surface The connections between these new types of integrals and the single, double, and triple integrals we have already met are given by the higherdimensional versions of the Fundamental Theorem of Calculus: • Greens Theorem • Stokes Theorem • Divergence Theorem Recall that we have learned about the following different type of functions function Domain D Range R (scalar) f ( t ) D ⊆ R R ⊆ R (vector) r ( t ) D ⊆ R R ⊆ V 2 or V 3 (scalar) f ( x,y ) D ⊆ R 2 R ⊆ R (scalar) f ( x,y,z ) D ⊆ R 3 R ⊆ R Today, we will introduce function Domain D Range R (vector field) F ( x,y ) D ⊆ R 2 R ⊆ V 2 (vector field) F ( x,y,z ) D ⊆ R 3 R ⊆ V 3 Vector field on R 2 Let D ⊆ R 2 . A vector field of on R 2 is a function F that assigns to each point ( x,y ) ∈ D a 2D vector F ( x,y ) . Vector field on R 3 Let D ⊆ R 3 . A vector field of on R 3 is a function F that assigns to each point ( x,y,z ) ∈ D a 3D vector F ( x,y,z ) . Suppose F is a vector function on R 2 . Since F ( x,y ) is a 2D vector, we can write F ( x,y ) in its component functions P and Q as follows: F ( x,y ) = P ( x,y ) i + Q ( x,y ) j = h P ( x,y ) ,Q ( x,y ) i or for short F = P i + Q j . Similarly, for a vector field F on R 3 , we can express F as F ( x,y,z ) = P ( x,y,z ) i + Q ( x,y,z ) j + R ( x,y,z ) k = h P ( x,y,z ) ,Q ( x,y,z ) ,R ( x,y,z ) i or for short F = P i + Q j + R k . Notice that P and Q are scalar functions of two variables. They are sometimes called scalar fields to distinguish them from vector fields. How can we visualize a vector field? The best way to picture a vector field F ( x,y ) is to draw the arrow representing the vector F ( x,y ) starting at the point ( x,y ) ....
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 Spring '08
 Kuchengyeaw
 Multivariable Calculus, Vector Calculus, lim, Vector field, Gradient

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