3 definition of the curl of vector fields in r 3 4

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3. Definition of the curl of vector fields in R 3 . 4. The vector differential operator (“del”). 5. div( -→ F ) = ∇ · -→ F , where -→ F is a vector filed. 6. curl( -→ F ) = ∇ × -→ F , where -→ F is a vector filed in R 3 . 10
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7. The curl is only defined for vector fields in R 3 . 8. Let f be a C 2 function in R 3 . Then curl( f ) = 0 . 9. Let -→ F be a vector field in R 3 . Then div(curl -→ F ) = 0 . 10. The physical interpretation of divergence. 11. Definition of a C 1 path. 12. Definition of a piecewise C 1 path. 13. Definition of simple curves. 14. The path integral of a real-valued function f along a C 1 path ~ c is equal to the area of the region along ~ c and below f . 15. The path integral of a real-valued function along a C 1 path ~ c : Z ~ c f ds = Z b a f ( ~ c ( t )) | -→ c 0 ( t ) | dt. 16. The path integral of a real-valued function along a piecewise C 1 path ~ c . 17. The path integral of a real-valued function along a path ~ c is independent of the parametrizations of ~ c . 14 Lecture 2/23 1. The path integral of a vector field ~ F along a C 1 path ~ c is equal to the work done by ~ F along ~ c . 11
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2. The path integral of a vector field ~ F along a C 1 path ~ c : Z ~ c ~ F · d~s = Z b a ~ F ( ~ c ( t )) · -→ c 0 ( t ) dt. 3. The path integral of a vector field along a piecewise C 1 path ~ c . 4. For a path integral of a vector field ~ F = ( F 1 , F 2 , F 3 ) in R 3 , we also use the notation: Z ~ c ~ F · d~s = Z ~ c F 1 dx + F 2 dy + F 3 dz. 5. The path integral of a vector field along a curve depends on the orienta- tions (parametrizations) of the curve. 6. Definition of a gradient vector field. 7. Definition of a conservative vector field. 8. The fundamental theorem for path integrals of gradient vector fields. 9. Let ~ F is a C 1 gradient vector field. If ~ c 1 and ~ c 2 are two oriented, simple curves with the same initial and terminal points, then Z ~ c 1 ~ F · d~s = Z ~ c 2 ~ F · d~s. In particular, if ~ c is a oriented, simple curves, then Z ~ c ~ F · d~s = 0 15 Lecture 2/28 1. The double integral of a function f ( x, y ) over the region R in R 2 is denoted by ZZ R f dA. 12
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2. Roughly speaking, the double integral of a function f ( x, y ) over the region R in R 2 is equal to the volume of the solid below the graph of f over the region R . 3. Definition of a rectangle in R 2 . 4. The definition of the double integral over a rectangle. 5. Let f be a continuous function defined on a rectangle R = [ a, b ] × [ c, d ] in R 2 . Then ZZ R f dA = Z b a Z d c f ( x, y ) dy dx = Z d c Z b a f ( x, y ) dx dy. 6. Definition of a type 1 region in R 2 . 7. Definition of a type 2 region in R 2 . 8. Definition of a type 3 region in R 2 . 9. Let D be a type 1 region given by D = { ( x, y ) | a x b, g 1 ( x ) y g 2 ( x ) } . Let f be a continuous function defined on D . The the double integral of f over D is given by ZZ D f dA = Z b a Z g 2 ( x ) g 1 ( x ) f ( x, y ) dy dx. 10. Let D be a type 2 region given by D = { ( x, y ) | c y d, h 1 ( y ) x h 2 ( y ) } .
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