Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
The composition of fx,y=x3y with x=r cost,y=r sint forms the function Fr,t=fxr,t,yr,t. Obtain the partial derivatives Fr and Ft by appropriate forms of the chain rule, and again by writing the rule for F explicitly. Show that the results agree.
An application of the chain rule gives
=fx xr+fy yr
=fx xt+fy yt
Writing Fr,t=fxr,t,yr,t=r4cos3tsint explicitly gives Fr=4⁢r3⁢cos⁡t3⁢sin⁡t and Ft=r44⁢cos⁡t2−3⁢cost2, in agreement with the chain-rule results.
Maple Solution - Interactive
Formal statement of the relevant chain rules
Context Panel: Differentiate≻With Respect To≻r
fxr,t,yr,t→differentiate w.r.t. rD1⁡f⁡x⁡r,t,y⁡r,t⁢∂∂r⁢x⁡r,t+D2⁡f⁡x⁡r,t,y⁡r,t⁢∂∂r⁢y⁡r,t
Context Panel: Differentiate≻With Respect To≻t
fxr,t,yr,t→differentiate w.r.t. tD1⁡f⁡x⁡r,t,y⁡r,t⁢∂∂t⁢x⁡r,t+D2⁡f⁡x⁡r,t,y⁡r,t⁢∂∂t⁢y⁡r,t
It is possible to obtain notational simplifications interactively, via the Typesetting Rules Assistant in the View menu. However, this is a tedious multistep process, so will not be pursued here.
Implement the chain rule
Context Panel: Assign Function
fx,y=x3y→assign as functionf
Context Panel: Assign Name
Calculus palette: Partial and ordinary differential operators
Press the Enter key.
Context Panel: Evaluate at a Point≻x=X,y=Y
Context Panel: Simplify≻Simplify
∂∂ x fx,y ⅆⅆ r X+∂∂ y fx,y ⅆⅆ r Y
→evaluate at point
∂∂ x fx,y ⅆⅆ t X+∂∂ y fx,y ⅆⅆ t Y
Obtain Fr and Ft from the explicit representation Fr,t=fxr,t,yr,t
Calculus palette: Partial differentiation operator
Context Panel: Evaluate and Display Inline
∂∂ r fX,Y = 4⁢r3⁢cos⁡t3⁢sin⁡t
∂∂ t fX,Y = −3⁢r4⁢cos⁡t2⁢sin⁡t2+r4⁢cos⁡t4= simplify 4⁢cos⁡t2−3⁢cos⁡t2⁢r4
Maple Solution - Coded
Simplified Maple notation is available if the commands to the right are first executed.
Although the chain rules for this problem could be written as Fr=fx xr+fy yr and Ft=fx xt+fy yt, Maple uses the D-operator notation to express the partial derivatives fx and fy, and cannot suppress the arguments of f once suppression of arguments has been applied to x and y.
Restore the variables x and y.
Define the function f.
Assign xr,t and yr,t to the names X and Y, respectively.
X≔r cost:Y≔r sint:
Apply the simplify and diff commands.
D1fX,Y diffX,r+D2fX,Y diffY,r
simplifyD1fX,Y diffX,t+D2fX,Y diffY,t
Obtain Fr and Ft from an explicit representation of Fr,t
Using the diff and simplify commands, explicitly differentiate fxr,t,yr,t.
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