Partial Derivatives of Functions (continued)                                        

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5.

 

 

 

 

 

 Physical Interpretation of the partial derivative   ∂f(x,y)/ ∂y

 

 Consider a surface    S   given by    z  =  f(x,y).   The intersection of the plane, PL,

 x  =  constant with the surface,  S,  defines the curve of intersection,  C.   Let the

 tangent line to  C  at point P with coordinates  (a, b, c) be  T.

               

 

Then the slope of the line tangent to the curve C at the point (a,b,c) in the y-direction is:

 

                     fy(x,y)  =    ∂f(x,y)/ ∂y      evaluated at the point P  (a,b,c)

 

Similarly one can imagine a plane, y = constant, intersecting the surface  z 

along a curve D (not shown).

 The slope of the line tangent to the curve D at the point (a,b,c) in the x-direction is:

 

                     fx(x,y)  =    ∂f(x,y)/ ∂x      evaluated at the point  P  (a,b,c)

 

 

Click here for an example.

 


Copyright © 2011 Richard C. Coddington

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