Partial differentiation. Background mathematics review

Transcription

1 Partial differentiation Background mathematics review David Miller

2 Partial differentiation Partial derivatives Background mathematics review David Miller

3 Partial derivative Suppose we have a function f(,) of two variables Then we can have the gradient of f as a function of with held constant f f which we write or sometimes just the partial derivative of f with respect to with constant partial d f b d - partial d

4 Partial derivative Suppose we have a function f(,) of two variables Then we can have the gradient of f as a function of with held constant f f which we write or sometimes just the partial derivative of f with respect to with constant partial d f b d - partial d

5 M Partial derivative f, We have a surface with height f, varing with the coordinates and At a specific position o, the height is f, o o o o o f, o o

6 Partial derivative f, M

7 Partial derivative f, Along the direction at point o, o the rate of change of height with slope at constant is the slope of the orange line f o M

8 Partial derivative f, M

9 M Partial derivative f,

10 M Partial derivative f, Along the direction at point o, o the rate of change of height with at constant is the slope of the orange line o slope f

11 M Second partial derivatives We can also form second (and higher order) partial derivatives f f is a second derivative and is a measure of the curvature of the function in the direction f,

12 M Second partial derivatives We can also form second (and higher order) partial derivatives f f is a second derivative and is a measure of the curvature of the function in the direction f,

13 M Second partial derivatives f, We can also form second (and higher order) partial derivatives f f is a second derivative and is a measure of the curvature of the function in the direction f f 0 0

14 MM Second partial derivatives We can also form second (and higher order) partial derivatives f f is a second derivative and is a measure of the curvature of the function in the direction f, f f 0 0

15 Second partial derivatives f, We can also form second (and higher order) partial derivatives f f is a second derivative and is a measure of the curvature of the function in the direction saddle point f f 0 0 MS

16 W Cross derivative The cross derivative f f is a measure of how warped a surface is Note that, for ordinar smooth functions f f f, f 0

17 Cross derivative The cross derivative f f is a measure of how warped a surface is Note that, for ordinar smooth functions f f f, f 0 W

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19 Partial differentiation Differentials Background mathematics review David Miller

20 Differential MP Look at one patch of a function f, long in the direction long in the direction How much does the height change, i.e., f, as we move from one corner to the other? f f,

21 Differential As we move b along f the height changes As we move b along f the height changes so the total change in height f f is f MP f f, f f

22 Differential In the limit as we make and ver small i.e., infinitesimal f f f becomes f f df d d which is called a differential or sometimes an eact differential

23 Total derivative We suppose we know the slopes of the hill in the two coordinate directions We presume we also know how fast we are moving in the and directions So, in some small time t we will have moved amounts and in the and directions v f f and d d and v dt dt d d t and t dt dt

24 Total derivative So, using the differential idea we have or or, in the limit of small t we have the total derivative f f f f d f d f t t dt dt f f d f d t dt dt df f d f d dt dt dt

25 Changing coordinates for partial derivatives Suppose we want the slopes of a hill f(,) along the South-East (a) and North-East (b) directions instead of along the East () and North () directions but we onl know the slopes along the East () and North () directions f f and North North-East b East a South-East

26 Changing coordinates for partial derivatives We do know that if we move South-East b one unit we move East b 1/ units since cos 451/ 1 i.e., a b North North-East b 1/ 45 1 East a South-East

27 Changing coordinates for partial derivatives We do know that if we move South-East b one unit we move East b 1/ units since cos 451/ 1 i.e., a b Similarl 1 a b North 1 45 North-East b 1 East a South-East

28 Changing coordinates for partial derivatives We do know that if we move South-East b one unit we move East b 1/ units since cos 451/ 1 i.e., a b Similarl 1 1 a b b a North 1 North-East b 45 1/ East a South-East

29 Changing coordinates for partial derivatives We do know that if we move South-East b one unit we move East b 1/ units since cos 451/ 1 i.e., a b Similarl 1 1 a b b b a a 1 North 1 45 North-East b 1 East a South-East

30 Changing coordinates for partial derivatives Suppose we make a small movement a along the a (South-East) direction and no movement along the b (North-East) direction 1 Then a a a b North a North-East b East a South-East

31 Changing coordinates for partial derivatives Suppose we make a small movement a along the a (South-East) direction and no movement along the b (North-East) direction 1 Then a a a b and 1 a a a b North a North-East b East a South-East

32 Changing coordinates for partial derivatives With these results 1 1 a a a a a b a b the resulting change f in the value of the function f(,) from this movement along the a (South- East) direction is f f f a a a a b b

33 Changing coordinates for partial derivatives Starting from dividing b a gives taking the limit of small a and noting that this is all done at constant b Since and are just numbers a b a b we can move them to get f f f a a a b a b f f f a a a b b f f f a a a b b b f f f a a a b b b

34 Changing coordinates for partial derivatives Since f f f a a a b b b holds for an function f(,) provided it is suitabl differentiable we can write more generall a b a b a b which is a general wa of changing the coordinates for a partial derivative

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