Calc1231W3T4 - Multivariable Chain Rule

Sunday, 5 August 2018

7:56 PM

Untitled picture.png Machine generated alternative text:
Recall the Folium of Descartes, which has Cartesian equation 
3 
— 34 + y 
—o 
with the rational parameterization 
x(t) — 
y(t) — 
1+t3 ' 
3t2 
1+t3 
The Folium is shown below (in black), along with the normal to the curve (in red) Through this parameterization the Folium can be regarded as a function oft. Let's find the derivative with 
respect to geometncally_ 
Ifwe let 
where C 
x 
.3xy+ y then the equation of the Folium is in the form 
O is constant The normal to the curve (shown in red) at the point [x,y] is given by the vector 
Dc 
Dy 

Untitled picture.png Machine generated alternative text:
Where, as a function of x and y: 
3Xh2 - 3Y 
Dc 
3Yh2 - 3X 
Dy 
We can think of the parameter t 
Where, as a function of t: 
dc 
dy 
dt 
as time; velocity (shown in blue) is the vector 
dt 
dc 
dy 
dt 
Geometrically the chain rule states that the derivative is the dot product: 
Of course, it's easy to see that this quantity equals zero since 
the normal and velocity vectors are perpendicular 
the function F(x, y) equals a constant; so the derivative must be zero
Untitled picture.png Machine generated alternative text:
Where, as a function of x and y: 
3Xh2 - 3Y 
Dc 
3Yh2 - 3X 
Dy 
We can think of the parameter t 
Where, as a function of t: 
dc 
dy 
dt 
as time; velocity (shown in blue) is the vector 
dt 
dc 
dy 
dt 
Geometrically the chain rule states that the derivative is the dot product: 
Of course, it's easy to see that this quantity equals zero since 
the normal and velocity vectors are perpendicular 
the function F(x, y) equals a constant; so the derivative must be zero 

Untitled picture.png Machine generated alternative text:
A plane in R 
has Cartesian equation 
If we let F(x, y, z) = 5x + 9y+ 12z then the partial derivatives are 
These are the components of the normal to the plane; which is the vector 
Now consider a surface in R3 with Cartesian equation 
Ifwe let = 5y + 4x + 3, the partial derivatives are 
loy 
= 17. 
= 17 
These are the components of VG the normal to the surface at the point [x, y, z]. In contrast to the plane, the normal to a general surface need not be constant.
Untitled picture.png Machine generated alternative text:
A plane in R 
has Cartesian equation 
If we let F(x, y, z) = 5x + 9y+ 12z then the partial derivatives are 
These are the components of the normal to the plane; which is the vector 
Now consider a surface in R3 with Cartesian equation 
Ifwe let = 5y + 4x + 3, the partial derivatives are 
loy 
= 17. 
= 17 
These are the components of VG the normal to the surface at the point [x, y, z]. In contrast to the plane, the normal to a general surface need not be constant. 

Untitled picture.png Machine generated alternative text:
Consider a function of three variables, such as 
Suppose x, y and z are functions of t. The chain rule states that 
dx DF dc 
dt ¯ Dc dt 
Of special importance are those functions where z is a function of the other variables, such as 
Dy dt 
dz 
Dz dt 
If we know x = cos(t) and y = sin(t), then z must be function of t too. We can find the derivative 
by first rewriting our equation in the form 
-2 
Then take the derivative of both sides with respect to t. The derivative of the right hand side is o 
O . The derivative of the left hand side requires the chain rule. 
First, we compute the normal: 
where; as a function of x and y: 
3 
Dc 
Dy 
Dz 
Then we compute the velocity: 
where as a function of t: 
dc 
-sin(t) 
dy 
cos(t) 
dt 
dx 
Dy 
Dz 
dt 
dy 
dz 
dt
Untitled picture.png Machine generated alternative text:
Consider a function of three variables, such as 
Suppose x, y and z are functions of t. The chain rule states that 
dx DF dc 
dt ¯ Dc dt 
Of special importance are those functions where z is a function of the other variables, such as 
Dy dt 
dz 
Dz dt 
If we know x = cos(t) and y = sin(t), then z must be function of t too. We can find the derivative 
by first rewriting our equation in the form 
-2 
Then take the derivative of both sides with respect to t. The derivative of the right hand side is o 
O . The derivative of the left hand side requires the chain rule. 
First, we compute the normal: 
where; as a function of x and y: 
3 
Dc 
Dy 
Dz 
Then we compute the velocity: 
where as a function of t: 
dc 
-sin(t) 
dy 
cos(t) 
dt 
dx 
Dy 
Dz 
dt 
dy 
dz 
dt 

Untitled picture.png Machine generated alternative text:
dz 
The value of — can be determined because it is the only unknown quantity in the equation 
dt 
dt 
So the answer can be expressed as a function of t: 
dz 

Untitled picture.png Machine generated alternative text:
Consider the curve 
dz 
as a function oft, where x = cos(t) and y = sin(t). 
Using the chain rule, or otherwise (at your peril); calculate 
dz 
dt 
Untitled picture.png Machine generated alternative text:
Consider the hyperbolic paraboloid z 
A drone is programmed to fly along the surface of the hyperbolic paraboloid. The flight instruments measure the velocity in the and z directions as 
dt 
dt 
dt 
The GPS measures x = 10 but the GPS does not measure y (to keep costs down). What is the y co-ordinate? The chain rule can tell us' It's 
97/8 
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Untitled picture.png Machine generated alternative text:
Consider the hyperbolic paraboloid z 
A drone is programmed to fly along the surface of the hyperbolic paraboloid. The flight instruments measure the velocity in the and z directions as 
dt 
dt 
dt 
The GPS measures x = 10 but the GPS does not measure y (to keep costs down). What is the y co-ordinate? The chain rule can tell us' It's 
97/8 

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