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
Ink Drawings

Ink Drawings

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

Ink Drawings

Ink Drawings
Ink Drawings
Ink Drawings
Ink Drawings

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