Introduction to Partial Differential Equations
Exercise Sheet 1
Ross Ogilvie 2nd September,
2024
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This exercise sheet is revision and does not count towards the exercise points. But please feel
free to attempt them (before or after the tutorial) and submit them for correction any
way.
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1.
- Chain rule in multiple variables Recall the chain rule for functions of
multivariable variables (Satz 10.4(iii) in Schmidt’s Analysis II script): Let
be differentiable
at and
be differentiable
at . Then
is differentiable
at and
|
-
(a)
- Why does this chain rule above use function composition, when the chain rule for functions of a single
variable uses multiplication? i.e.
|
-
(b)
- Suppose that
and .
Express the chain rule with partial derivatives to show that
|
-
(c)
- Write the above formula in terms of gradients and dot products.
-
(d)
- Consider the function
and the polar coordinates .
Compute the radial and angular derivatives of .
-
(e)
- Consider a scalar function
of
variables and a function .
Write an expression for the derivative of
with respect to .
-
2.
- Contour Diagrams
Consider the function
defined by .
-
(a)
- Use computer assistance to draw a contour diagram for this function. A contour is another word
for a level set .
For example https://www.desmos.com/calculator/8equb62lyq.
-
(b)
- What is the maximum and minimum of this function? Where are its critical points? (Give
approximate values.)
-
3.
- Multiindices and the Generalised Leibniz rule In this question we introduce multiindex notation. A
multiindex of
variables is a vector .
-
(a)
- Let
be coordinates on .
Write out the full expression for the derivative .
-
(b)
- Why do we need to assume that partial derivatives commute for multiindex notation to be
useful?
-
(c)
- Which multiindices satisfy
and which satisfy ?