Vectors Calaulus 1.12

Partial Derivatives Definition of a Partial Derivative Let f(x,y) be a function of two variables.  Then we define the partial derivatives  as     Definition of the Partial Derivative          if these limits exist.   Algebraically, we can think of the partial derivative of a function with respect to x as the derivative of the function with y held constant.  Geometrically, the derivative with respect to x at a point P represents the slope of the curve that passes through P whose projection onto the xy plane is a horizontal line.  (If you travel due East, how steep are you climbing?) Example Let         f(x,y) = 2x + 3y then         We also use the notation fx  and fy for the partial derivatives with respect to x and y  respectively. Exercise: Find fy for the function from the example above. Finding Partial Derivatives the Easy Way Since a partial derivative with respect to x is a derivative with the rest of the variables held constant, we can find the partial derivative by taking the regular derivative considering the rest  of the variables as constants. Example   Let         f(x,y)  =  3xy2 - 2x2y then         fx  =  3y2 - 4xy and         fy  =  6xy - 2x2 Exercises  Find both partial derivatives for f(x,y) = xy sin x                 x + y f(x,y) =                                     x - y Higher Order Partials Just as with function of one variable, we can define second derivatives for functions of two variables.  For functions of two variables, we have four types:         fxx,     fxy     fyx     and     fyy Example Let         f(x,y)  =  y ex then         fx  =  yex and         fy  =  ex Now taking the partials of each of these we get:         fxx = y ex        fxy = ex        fyx = ex       and       fyy = 0 Notice that            fxy  =   fyx                                      Theorem Let f(x,y) be a function with continuous second order derivatives, then               fxy  =   fyx   Functions of More Than Two Variables Suppose that         f(x,y,z)  =  xy - 2yz  is a function of three variables, then we can define the partial derivatives in much the same way as we defined the partial derivatives for three variables.   We have         fx = y         fy = x - 2z      and       fz = -2y Application:     The Heat Equation Suppose that a building has a door open during a snowy day.  It can be shown that the equation         Ht  =  c2Hxx      models this situation where H is the heat of the room at the point x feet away from the door at time t.   Show that         H = e-t cos(x/c) satisfies this differential equation. Solution We have         Ht  =  -e-t cos(x/c)         Hx  =  -1/c e-t sin(x/c)         Hxx  =  -1/c2 e-t cos(x/c) So that         c2Hxx  =  -e-t cos(x/c) And the result follows.