Chapter 4: Partial Differentiation
Section 4.3: Chain Rule
The composition of f⁡x,y,z=x2+y2+z2 with x⁡r,s=ⅇr⁢cos⁡s, y⁡r,s=ⅇr⁢sin⁡s, z⁡r,s=r⁢s forms the function Fr,s=fxr,s,yr,s,zr,s. Obtain the partial derivatives Fr and Fs 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+fz zr
=ⅇ2 r⁢cos2sⅇ2 r⁢cos2s+ⅇ2 r⁢sin2s+r2⁢s2+ⅇ2 r⁢sin2sⅇ2 r⁢cos2s+ⅇ2 r⁢sin2s+r2⁢s2+r⁢s2ⅇ2 r⁢cos2s+ⅇ2 r⁢sin2s+r2⁢s2
=fx xs+fy ys+fz zs
Writing Fr,s=fxr,s,yr,s,zr,s=ⅇ2⁢r+r2⁢s2 explicitly gives Fr=ⅇ2⁢r+r⁢s2ⅇ2⁢r+r2⁢s2 and Fs=r2⁢sⅇ2⁢r+r2⁢s2, 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,s,yr,s,zr,s→differentiate w.r.t. rD1⁡f⁡x⁡r,s,y⁡r,s,z⁡r,s⁢∂∂r⁢x⁡r,s+D2⁡f⁡x⁡r,s,y⁡r,s,z⁡r,s⁢∂∂r⁢y⁡r,s+D3⁡f⁡x⁡r,s,y⁡r,s,z⁡r,s⁢∂∂r⁢z⁡r,s
Context Panel: Differentiate≻With Respect To≻t
fxr,s,yr,s,zr,s→differentiate w.r.t. sD1⁡f⁡x⁡r,s,y⁡r,s,z⁡r,s⁢∂∂s⁢x⁡r,s+D2⁡f⁡x⁡r,s,y⁡r,s,z⁡r,s⁢∂∂s⁢y⁡r,s+D3⁡f⁡x⁡r,s,y⁡r,s,z⁡r,s⁢∂∂s⁢z⁡r,s
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.
Be sure to use Maple's exponential "e" when writing the functions X and Y.
Implement the chain rule
Context Panel: Assign Function
fx,y,z=x2+y2+z2→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,z=Z
Context Panel: Simplify≻Simplify
∂∂ x fx,y,z ⅆⅆ r X+∂∂ y fx,y,z ⅆⅆ r Y+∂∂ z fx,y,z ⅆⅆ r Z
→evaluate at point
∂∂ x fx,y,z ⅆⅆ s X+∂∂ y fx,y,z ⅆⅆ s Y+∂∂ z fx,y,z ⅆⅆ s Z
Obtain Fr and Fs from the explicit representation Fr,s=fxr,s,yr,s,zr,s
Calculus palette: Partial differentiation operator
Context Panel: Evaluate and Display Inline
∂∂ r fX,Y,Z = 12⁢2⁢ⅇr2⁢cos⁡s2+2⁢ⅇr2⁢sin⁡s2+2⁢r⁢s2ⅇr2⁢cos⁡s2+ⅇr2⁢sin⁡s2+r2⁢s2= simplify ⅇ2⁢r+r⁢s2ⅇ2⁢r+r2⁢s2
∂∂ s fX,Y,Z = r2⁢sⅇr2⁢cos⁡s2+ⅇr2⁢sin⁡s2+r2⁢s2= simplify r2⁢sⅇ2⁢r+r2⁢s2
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+fz zr and Fs=fx xs+fy ys+fz zs, Maple uses the D-operator notation to express the partial derivatives fx, fy, and fz 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,s and yr,s to the names X and Y, respectively.
X≔ⅇr coss:Y≔ⅇr sins:Z≔r s:
Apply the simplify and diff commands.
simplifyD1fX,Y,Z diffX,r+D2fX,Y,Z diffY,r+D3fX,Y,Z diffZ,r
simplifyD1fX,Y,Z diffX,s+D2fX,Y,Z diffY,s+D3fX,Y,Z diffZ,s
Obtain Fr and Fs from an explicit representation of Fr,s
Using the diff and simplify commands, explicitly differentiate fxr,s,yr,s,zr,s.
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