math226_summary13

# math226_summary13 - 1 Gradient Divergence Curl(13.1 13.5 a...

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1. Gradient, Divergence, Curl (13.1, 13.5) a) Given a scalar function f , its gradient is the vector ﬁeld grad f = f = h ∂f/∂x,∂f/∂y,∂f/∂z i = h f x ,f y ,f z i ; b) Given a vector ﬁeld ~ F = h P,Q,R i , its divergence is the scalar function div ~ F = ∇ · ~ F = ∂P/∂x + ∂Q/∂y + ∂R/∂z. c) Given a vector ﬁeld ~ F = h P,Q,R i , its curl is the vector ﬁeld curl ~ F = ∇ × ~ F = h ∂R/∂y - ∂Q/∂z,∂P/∂z - ∂R/∂x,∂Q/∂x - ∂P/∂y i . 2. Line Integrals (13.2) Given ~ F = h P,Q,R i and a curve C : ~ r ( t ) = h x ( t ) ,y ( t ) ,z ( t ) i , a t b, Z C ~ F · d~ r = Z C Pdx + Qdy + Rdz = Z b a ~ F ( ~ r ( t )) · ~ r 0 ( t ) dt = Z b a ( Px 0 + Qy 0 + Rz 0 ) dt. Remark 1. In 2D, the curve C : y = g ( x ) ,a x b, can be parametrized as C : ~ r ( t ) = h t,g ( t ) i , a t b. PROPERTIES (13.2, 13.3). 1. R C ~ F · d~ r does not depend on orientation preserving parametrization. 2.

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math226_summary13 - 1 Gradient Divergence Curl(13.1 13.5 a...

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