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1.2 Differential Calculus

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.

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1.2 Differential Calculus

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  1. 1.2 Differential Calculus

  2. 1.2.1 The Gradient

  3. 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:

  4. Example: Multiplication by a scalar-the gradient. The order of the factors does matter! 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.

  5. 1.2.4 The Divergence divergence of a vector field

  6. large divergence no divergence

  7. large divergence

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

  9. 1.2.5 The Curl Curl of the a vector field Keep track of the order in evaluating the determinant!

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

  11. (x-y,x+y).

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

  13. 1.2.6 Product Rules • Apply 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.

  14. 1.2.7 Second derivatives

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