Vector field definition let d be a set in r 2 a plane

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vector field . Definition Let D be a set in R 2 (a plane region). A vector field on R 2 is a function F that assigns to each point ( x , y ) in D a two-dimensional vector F ( x , y ).
The next definition describes the notion of a vector field . Definition Let D be a set in R 2 (a plane region). A vector field on R 2 is a function F that assigns to each point ( x , y ) in D a two-dimensional vector F ( x , y ). Example We have already seen an example of a vector field associated to a function f ( x , y ) defined on a domain D R 2 , namely the gradient vector field f ( x , y ) = h f x ( x , y ) , f y ( x , y ) i . In nature and in physics, we have the familiar examples of the velocity vector field in weather and force vector fields that arise in gravitational fields, electric and magnetic fields.
Ocean currents off Nova Scotia
A vector field in the plane together with its integral curves.
Definition Let E be a subset of R 3 . A vector field on R 3 is a function F that assigns to each point ( x , y , z ) in E a three-dimensional vector F ( x , y , z ).
Definition Let E be a subset of R 3 . A vector field on R 3 is a function F that assigns to each point ( x , y , z ) in E a three-dimensional vector F ( x , y , z ). Note that a vector field F on R 3 can be expressed by its component functions. So if F = h P , Q , R i , then: F ( x , y , z ) = P ( x , y , z ) i + Q ( x , y , z ) j + R ( x , y , z ) k .
We now describe our first kind of ”line integral”. These type of integrals arise from integrating a function along a curve C in the plane or in R 3 . The types of line integral described in the next definition is called a line integral with respect to arc length .
We now describe our first kind of ”line integral”. These type of integrals arise from integrating a function along a curve C in the plane or in R 3 . The types of line integral described in the next definition is called a line integral with respect to arc length . Definition Let C be a smooth curve in R 2 . Given n , consider n equal subdivisions of lengths Δ i s ; let ( x * i , y * i ) denote the midpoints of the i -th subdivision. If f is a real valued function defined on C , then the line integral of f along C is Z C f ( x , y ) ds = lim n →∞ n X j =1 f ( x * i , y * i ) Δ s i if this limit exists.
Interpretation of the line integral R C f ( x , y ) ds as being the area under the graph of f over C
The following formula can be used to evaluate a line integral. Theorem Suppose f ( x , y ) is a continuous function on a differentiable curve C ( t ), C : [ a , b ] R 2 . Then Z C f ( x , y ) ds = Z b a f ( x ( t ) , y ( t )) s dx dt 2 + dy dt 2 dt . Note that in the above formula, s dx dt 2 + dy dt 2 , is the speed of C ( t ) at time t .
Example Evaluate R C 2 x ds , where C consists of the arc C of the parabola y = x 2 from (0 , 0) to (1 , 1).
Example Evaluate R C 2 x ds , where C consists of the arc C of the parabola y = x 2 from (0 , 0) to (1 , 1). Solution: We can choose x as the parameter and the equations for C become x = x y = x 2 0 x 1 . Therefore Z C 2 x ds = Z 1 0 2 x s dx dx 2 + dy dx 2 dx = Z 1 0 2 x p 1 + 4 x 2 dx = 1 4 · 2 3 (1 + 4 x 2 ) 3 2 1 0 = 5 5 - 1 6 .
One can also define in a similar manner the line integral of a function f along a curve C in R 3 .

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