Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
The composition of fx,y with x=r cost, y=r sint produces the function Fr,t=fxr,t,yr,t. Express fx and fy in terms of Fr and Ft.
Start with the identity fr cost,r sint=Fr,t and form two equations by differentiating first with respect to r, then with respect to t. The resulting equations are
fx cost+fy sint=Fr and fx −r sint+fy r cost=Ft
Solve these equations for fx and fy. One approach to solving these equations "by hand" is to write the system in matrix form and apply Cramer's rule. Thus, write
cos(t)sin(t)−r sin(t)r cos(t)fxfy=FrFt
fx=FrsintFtr costcostsint−r sintr cost=Fr r cost−Ft sintr cos2t+sin2t=Fr cost−Ft sint/r
fy=costFr−r sintFtcostsint−r sintr cost=Fr r sint+Ft costr cos2t+sin2t=Fr sint+Ft cost/r
are the desired solutions.
Maple Solution - Interactive
Assign the identity f→F the name q
Context Panel: Assign to a Name≻q
fr cost,r sint=Fr,t→assign to a nameq
Differentiate and solve for fx and fy
Calculus palette: Partial-derivative operator
Context Panel: Solve≻Solve for Variables≻fx and fy
(See Figure 4.3.29(a) for data entry.)
Context Panel: Simplify≻Simplify
Figure 4.3.29(a) Entering fx and fy
∂∂ r q,∂∂ t q
Extract from this display the solution
fx=Fr cost−Ft sint/r and fy=Fr sint+Ft cost/r
Maple Solution - Coded
These commands allow xr,t and yr,t to be represented by x and y,respectively; and for derivatives to be written with subscripts.
Differentiate the equation fx,y=Fr,t with respect to r and t
Apply the diff command.
Solve the two equations for fx and fy
Apply the solve command.
Display the two solutions on separate lines
Invoke the print command, using the tilde (~) to apply the command to each item in q1, the set of solutions.
Replace the names x and y with r cost and r sint, respectively; display the results
Apply the eval, simplify, and expand commands to obtain the results.
Invoke the print command, using the tilde (~) to apply the command to each item in q2.
q2≔expandsimplifyevalq1,x=r cost,y=r sint:
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