Chain rule for functions of 2, 3 variables (Sect. 14.4 ...
Chain rule for functions of 2, 3 variables (Sect. 14.4)
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Review: Chain rule for f : D ? R R.
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Functions of two variables, f : D ? R2 R.
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Chain rule for functions defined on a curve in a plane.
Chain rule for change of coordinates in a plane.
Functions of three variables, f : D ? R3 R.
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Chain rule for change of coordinates in a line.
Chain rule for functions defined on a curve in space.
Chain rule for functions defined on surfaces in space.
Chain rule for change of coordinates in space.
A formula for implicit differentiation.
Review: The chain rule for f : D ? R R
Chain rule for change of coordinates in a line.
Theorem
If the functions f : [x0 , x1 ] R and x : [t0 , t1 ] [x0 , x1 ] are
differentiable, then the function
f? : [t0 , t1 ] R given by the
composition f?(t) = f x(t) is differentiable and
dx
d f?
df
(t) =
x(t)
(t).
dt
dx
dt
Notation:
d f?
df dx
=
.
dt
dx dt
Alternative notations are f?0 (t) = f 0 x(t) x 0 (t) and f?0 = f 0 x 0 .
The equation above is usually written as
Review: The chain rule for f : D ? R R
Chain rule for change of coordinates in a line.
Example
The volume V of a gas balloon depends on the temperature F in
Fahrenheit as V (F ) = k F 2 + V0 . Let F (C ) = (9/5)C + 32 be the
temperature in Fahrenheit corresponding to C in Celsius. Find the
rate of change V? 0 (C ).
Solution: Use the chain rule to derivate V? (C ) = V (F (C )),
9
9
0
0
0
0
C + 32 .
V? (C ) = V (F ) F = 2k F F = 2k
5
5
9
18k
C + 32 .
We conclude that V 0 (C ) =
C
5
5
9
2
Remark: One could first compute V? (C ) = k
C + 32 + V0
9 5 9
and then find the derivative V? 0 (C ) = 2k
C + 32 .
5
5
Chain rule for functions of 2, 3 variables (Sect. 14.4)
I
Review: Chain rule for f : D ? R R.
I
I
Functions of two variables, f : D ? R2 R.
I
I
I
Chain rule for functions defined on a curve in a plane.
Chain rule for change of coordinates in a plane.
Functions of three variables, f : D ? R3 R.
I
I
I
I
Chain rule for change of coordinates in a line.
Chain rule for functions defined on a curve in space.
Chain rule for functions defined on surfaces in space.
Chain rule for change of coordinates in space.
A formula for implicit differentiation.
Functions of two variables, f : D ? R2 R
The chain rule for functions defined on a curve in a plane.
Theorem
If the functions f : D ? R2 R and r : R D ? R2 are
differentiable, with r(t) = hx(t), y (t)i, then the function
f? : R R given by the composition f?(t) = f r(t) is
differentiable and holds
dx
dy
?f
?f
d f?
r(t)
r(t)
(t) =
(t) +
(t).
dt
?x
dt
?y
dt
Notation:
The equation above is usually written as
d f?
?f dx
?f dy
=
+
.
dt
?x dt
?y dt
An alternative notation is f?0 = fx x 0 + fy y 0 .
Functions of two variables, f : D ? R2 R.
The chain rule for functions defined on a curve in a plane.
Example
Find the rate of change of the function f (x, y ) = x 2 + 2y 3 , along
the curve r(t) = hx(t), y (t)i = hsin(t), cos(2t)i.
Solution: The rate of change of f along the curve r(t) is the
derivative of f?(t) = f (r(t)) = f (x(t), y (t)). We do not need to
compute f?(t) = f (r(t)). Instead, the chain rule implies
f?0 (t) = fx (r) x 0 + fy (r) y 0 = 2x x 0 + 6y 2 y 0 .
Since x(t) = sin(t) and y (t) = cos(2t),
f?0 (t) = 2 sin(t) cos(t) + 6 cos2 (2t) ?2 sin(2t) .
The result is f?0 (t) = 2 sin(t) cos(t) ? 12 cos2 (2t) sin(2t).
C
Functions of two variables, f : D ? R2 R
The chain rule for change of coordinates in a plane.
Theorem
If the functions f : R2 R and the change of coordinate functions
x, y : R2 R are differentiable, with x(t, s) and y (t, s), then the
function f? : R2 R given by the composition
f?(t, s) = f x(t, s), y (t, s) is differentiable and holds
f?t = fx xt + fy yt
f?s = fx xs + fy ys .
Remark: We denote by f (x, y ) the function values in the
coordinates (x, y ), while we denote by f?(t, s) are the function
values in the coordinates (t, s).
Functions of two variables, f : D ? R2 R
The chain rule for change of coordinates in a plane.
Example
Given the function f (x, y ) = x 2 + 3y 2 , in Cartesian coordinates
(x, y ), find the derivatives of f in polar coordinates (r , ).
Solution: The relation between Cartesian and polar coordinates is
x(r , ) = r cos(),
y (r , ) = r sin().
The function f in polar coordinates is f?(r , ) = f (x(r , ), y (r , )).
The chain rule says f?r = fx xr + fy yr and f? = fx x + fy y , hence
f?r = 2x cos() + 6y sin() ? f?r = 2r cos2 () + 6r sin2 ().
f? = ?2xr sin() + 6yr cos(),
f? = ?2r 2 cos() sin() + 6r 2 cos() sin().
C
Chain rule for functions of 2, 3 variables (Sect. 14.4)
I
Review: Chain rule for f : D ? R R.
I
I
Functions of two variables, f : D ? R2 R.
I
I
I
Chain rule for functions defined on a curve in a plane.
Chain rule for change of coordinates in a plane.
Functions of three variables, f : D ? R3 R.
I
I
I
I
Chain rule for change of coordinates in a line.
Chain rule for functions defined on a curve in space.
Chain rule for functions defined on surfaces in space.
Chain rule for change of coordinates in space.
A formula for implicit differentiation.
Functions of three variables, f : D ? R3 R.
Chain rule for functions defined on a curve in space.
Theorem
If the functions f : D ? R3 R and r : R D ? R3 are
differentiable, with r(t) = hx(t), y (t), z(t)i, then the
function
?
?
f : R R given by the composition f (t) = f r(t) is
differentiable and holds
d f?
?f dx
?f dy
?f dz
=
+
+
.
dt
?x dt
?y dt
?z dt
Notation:
The equation above is usually written as
f?0 = fx x 0 + fy y 0 + fz z 0 .
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