20-math - Mathematics Review Applied Computational Fluid...

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1 Mathematics Review Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org © André Bakker (2002-2006)
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2 Vector Notation - 1 = = . = x = i i y x Y X product: ("inner") Scalar = r r Z Y X product: ("outer") Vector r r r = × j i ij y x a A Y X product tensor Dyadic = = r r r r : ) " (" i i fx y Y f X tion: Multiplica = = r v ji ij b a B A product: ("inner") dot Double = r r r r : = :
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3 Vector Notation - 2 = . = . k ik i x a y Y X A product: Scalar = = r r r r kj ik ij b a c C B A product: Scalar = = r r r r r r = B a F F B A product: ker Kronec ij ij r r r r r r r r = =
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4 The gradient of a scalar f is a vector: Similarly, the gradient of a vector is a second order tensor, and the gradient of a second order tensor is a third order tensor. The divergence of a vector is a scalar: Gradients and divergence k j i z f y f x f f f grad + + = = z A y A x A div z y x + + = = .A A = = .
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5 The curl (“rotation”) of a vector v ( u,v,w ) is another vector: Curl - - - = × = = y u x v x w z u z v y w rot curl , , v v v = x
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6 Definitions - rectangular coordinates k j i A - + - + - = × y A x A x A z A z A y A x y z x y z 2 2 2 2 2 2 2 z f y f x f f + + = k j i A z y x A A A 2 2 2 2 = k j i z f y f x f f + + = z A y A x A z y x + + = .A
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7 Identities f g g f g f = ) ( B) ( A A) ( B )B (A )A (B B) (A × × × × = ) ( ) ( ) ( A A A = f f f B) ( A A B B A × - × = × ) ( ) ( ) ( ) ( ) ( A A A × × = × f f f A)B ( B)A ( )B (A )A (B B) (A - - = × ×
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8 Identities A A) ( A 2 - = × × k j i )B (A + + + + + + + + = z B A y B A x B A z B A y B A x B A z B A y B A x B A z z z y z x y z y y y x x z x y x x τ volume bounds which surface the is S where d f d f S S × - = × = da A A da τ τ τ τ ) (
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9 Differentiation rules
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10 Integral theorems τ volume bounds which surface the is S where d d theorem divergence Gauss S ∫∫∫ ∫∫ = τ τ A s A : ' ) . . ( : ' volume a bound not does but surface a be may S e i S surface open the bounds which curve closed the is C where d d theorem Stokes S C D 3 - × = ∫∫ s A) ( l A
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11 Euclidian Norm Various definitions exist for the norm of vectors and matrices. The most well known is the Euclidian norm. The Euclidian norm || V || of a vector V is: The Euclidian norm || A || of a matrix A is: Other norms are the spectral norm, which is the maximum of the individual elements, or the Hölder norm, which is similar to the Euclidian norm, but uses exponents p and 1/p instead of 2 and 1/2, with p a number larger or equal to 1.
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