MATH224: Studio 5 Assignment (Partial Derivatives, Directional Derivatives and the
Mulivariate Chain Rule)
Created by: Gus Greivel
Last Modified: 9/24/2015, 8:32am
Background:
A differential equation, often called an ordinary differential equations or ODE ( Wikipedia –
Ordinary Differential Equation ), is an equation involving a function of a single variable and some set of that function’s derivatives. Several physical, economic and biological models studied in the core curriculum at CSM are based on the formulation of an ODE from fundamental scientific laws and principles. The solution to an ODE is a function (or family of functions) that satisfy the equation when differentiated. You will all take a course in ODEs where you will learn various techniques to solve these equations and analyze their behaviors.
A partial differential equation or PDE ( Wikipedia – Partial Differential Equation ) is an equation that involves an unknown function of two or more variables and partial derivatives of that unknown function. Many physical models that you will study in the context of your upperdivision coursework are based on the formulation of a PDE. In general, there are many different functions that may satisfy a particular PDE and finding or approximating solutions can be quite challenging. If you find them interesting (and we think you should), we do offer an elective course on PDEs.
In this studio activity, we will consider one particular PDE as well as one possible solution.
Specifically, we will consider the heat equation in 2-D ( Wikipedia – Heat Equation ) and a function that appears to make sense in terms of the expected behavior of the model…
In a study of frost penetration it was found that the temperature
T
at time t
(measured in days) at a depth x
(measured in feet) can be modeled by the function
T ( x , t )

T
0

T
1 e
  x sin
  t
  x
 where w and  are positive constants and
T
0
and
T
1
are constants.
Calculus III Practice Question 1:
(a) Find the range of this function. What do you think the constants
T
0 represent physically?
and
T
1
(b) Find


T x
and


T t
. What do you think these quantities represent physically?
(c) Find all of the second partials of
T ( x , t )
. Can you verify Clairaut’s Theorem for this function?
(d) If we choose the following values of the constants:
T
0
=
0, T
1
=
10, w =
1, l =
1
, this function reduces to the equivalent of f ( x , y )
=
10 e
x of the tangent plane to the graph of z
= f ( x , y ) sin( y
x )
. Find the equation
at the point
(
0, p
4
)
and then use
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Mathematica to graph the function and this tangent plane near the point of tangency.
(e) Show that the original function
T
satisfies the heat equation
T t
 kT xx
for a certain constant k
. What is k
?
Multivariate Chain Rule:
Based on our in-class lectures on the Multivariate Chain Rule this week, please attempt the following problems:
Calculus III Practice Question 2:
Consider the equation x
2
+ y
2
+ z
2
= e xyz . (This equation cannot be solved for z as an explicit function of x and y.)
(a) Use the implicit differentiation result from the multivariate chain rule to find
¶ z
¶ x and
¶ z
¶ z
¶ z
¶ y
for this equation. Are there any points where
¶ x
and/or
¶ y
won’t exist?
What is true about points where
¶ z
¶ z
¶ x
and/or
¶ y
don’t exist in general?
(b) Explain how you would go about finding the equation of the plane tangent to the surface defined as by the graph of this equation at any given point where
¶ z
¶ x
and
¶ z
are defined.
¶ y
Directional Derivatives:
Based on our in-class lectures on the Directional Derivative and the Gradient this week, please attempt the following problems:
Calculus III Practice Question 3:
Consider the function f ( x , y , z )
= x
2 -
2 xy
+ yz
at the point
(1,
-
1, 2)
.
(a) Find the instantaneous rate of change in f at this point in the direction of the vector .
y
2 + z
2
(b) Find the maximum rate of change in f at this point and the direction in which it occurs. How would you justify your answers using the geometry of the definition of the dot product?
2

Reflection questions:
(a) What is your perspective of the relationship between (i) the multivariate chain rule and the chain rule you studied in Calculus I and (ii) the gradient of a function of 2 or more variables and the derivative of a function of 1 variable that you studied in Calculus I?
(b) Please comment on how these results from Calculus III extend or generalize the results from Calculus I.
Multivariate Chain Rule (Challenge Question): While we are on the topic of PDE and the chain rule, it makes sense to consider how the problem changes on non-Cartesian geometries.
The heat equation discussed above is a PDE of the form,
𝑇 𝑡
= 𝑘 𝑇 𝑥𝑥
As the spatial dimension increases, the right-hand side sees an increase in second-order spatial derivatives and becomes
𝑇 𝑡
= 𝑘( 𝑇 𝑥𝑥
+ 𝑇 𝑦𝑦
) which models the flow of heat in a thin rectangular plate. If this plate were circular then we would want to transform the Cartesian coordinates to polar coordinates. In this case, the righthand side must also be converted to polar coordinates, which is a job for the chain rule where 𝑟, 𝜃 are intermediate variables and 𝑥, 𝑦 are the independent variables. The work to convert the right-hand side of this 2-D, in space, PDE is outlined in this Onenote notebook.
For 3-D in space,
𝑇 𝑡
= 𝑘( 𝑇 𝑥𝑥
+ 𝑇 𝑦𝑦
+ 𝑇 𝑧𝑧
) one often considers spherical coordinates. Mathworld has the relevant results in Eq. (1)-(6) and (90)-(92).
Your challenge, if you choose to accept it, is to derive Eq. (92). A submission site will be created and extra-credit points will be awarded for clear, correct and concise work submitted by any group or individual.
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