Ch4 - Ch.4 PHYS2004 Quantitative Methods for Basic Physics...

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Ch.4 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 1 Chapter 4 Functions of Two or More Variables 4.1 Partial Derivatives 4.1.1 Introduction If  yfx , dy dx = the slope of the curve   ; OR = the rate of change of y with respect to x . If , zfx y , All the points of the above equation form a surface in 3D-space. Suppose x is constant, a plane x = const. intersecting the surface to give a curve . zy  = the slope of the curve . = the rate of change of z with respect to y when x is kept constant. Similarly, a plane y = const. intersecting the surface to give a curve. zx = the slope of the curve . = the rate of change of z with respect to x when y is kept constant. Some Notations Given   , y , 1 xx fz f zf  , 2 yy f  ,
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Ch.4 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 2 22 21 yx yx ffz f zf x y xy     , 11 xx xx f xx x x , 233 111 z xxx xxx ff f x x , 23 3 211 y yxx yxx z f xy xy     . We can also consider function of more variables than two, although in this case it is not so easy to give a geometrical interpretation. For example, the temperature T of the air in a room might depend on the point   ,, xyz at which we measured it and on the time t ; we would write   ,,, TTx y z t . Ty = the rate at which T is changing with y for fixed x and z at one instant of time t . A notation which is frequently used in applications ( particularly thermodynamics ) is y z x    , meaning zx when z is expressed as a function of x and y . (Note two different uses of the subscript y: y f meant f y . A subscript on a partial derivative, however, does not mean another derivative, but just indicates the variable being constant in the indicated partial differentiation.) Example: Let zx y  . By introducing polar coordinates r and , z may also be expressed as:   2 2 2 cos sin cos sin zr r r   , (4.1.1a) 222 x y x r , ( 4 . 1 . 1 b ) 2 y r y  . ( 4 . 1 . 1 c )
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Ch.4 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 3 For each new expression, let us find zr . For (4.1.1a):  22 2c o s s i n z r r     , (4.1.2a) For (4.1.1b): 2 x z r r , ( 4 . 1 . 2 b ) For (4.1.1c): 2 y z r r . ( 4 . 1 . 2 c ) These three expressions for  have different values and are derivatives of three different functions, so we distinguish them as indicated by writing the second independent variable as a subscript.
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Ch4 - Ch.4 PHYS2004 Quantitative Methods for Basic Physics...

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