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7. Approximations_for_partial_derivatives_c

7. Approximations_for_partial_derivatives_c - 2.3...

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2.3 Approximations for Partial Derivatives Extension of the ideas developed in section 2.2 for partial derivatives is straightforward. Just keep in mind the physical/geometrical meaning of partial derivatives. Consider a function of two variables: ) , ( y x f z = Discrete representation: - set up a grid of ) , ( i i y x values in the ) , ( y x plane - i x values, x Δ apart and j y values, y Δ apart - x Δ and y Δ spacing can be the same or can be different - choose spacing to resolve the shape of the ) , ( y x f surface Recall notation for values of the dependent variable at each (x,y) grid point: . ) , ( ) , ( ) , ( 1 , 1 , 1 1 , etc f y x f f y x f f y x f j i j i j i j i j i j i + + + + = = = The physical interpretation of the partial derivative: x) fixed a for (i.e., given x a for y with y) f(x, of Change of Rate y) fixed a for (i.e., y given a for with x y) f(x, of Change of Rate = = y f x f The geometrical interpretation of the partial derivative: y of direction in the y) f(x, of Slope x of direction in the y) f(x, of Slope = = y f x f x y j y j+1 y j-1 x
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