1.2 Differential Calculus

1.2.1 The Gradient

yT(x,y)=const.xgradient:

1.2.3 The del Operator The del is similar to a vector, but it is an operator.It acts on (takes the partial derivatives of)everything to its right.

1.2.4 The Divergencedivergence of a vector field

no divergence

large divergence

F(x,y) = (2*x,-y). F(x,y) = (2*x,-y)-(6,-1).

1.2.5 The CurlCurl of the a vector field Keep track of the order in evaluating the determinant!

G(x,y) - G(x0,y0) (-y-3,x-3) (-y+3,x+3)

(x-y,x+y).

Turbulent motion of air around a vibrating cylinder in a wind tunnel.

1.2.6 Product RulesApply del to all factors.Keep track of the type of multiplication (dot vs. cross, how connected).Arrange in standard form (gradient, curl, divergence).Most important products are listed in the book.

1.2.7 Second derivatives