1.2 Differential Calculus

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1.2 Differential Calculus. 1.2.1 The Gradient. points in the direction of maximum increase of T. It is perpendicular to the surface T=const. gives the slope (rate of increase) along this direction. T(x,y)=const. θ. y. x. gradient:. Example: Multiplication by a scalar-the gradient. - PowerPoint PPT Presentation

Transcript of 1.2 Differential Calculus

  • 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