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Written Homework 8
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FTC: |
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+ |
Product Rule: |
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= |
Integration by Parts: |
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Using the divergence theorem,
Problem 1
is a function,
is a function, and
is a vector field.
Part A
We need to verify this.
Let
, so
Part B
Find
Plug in
:
, where
is the Laplace operator
Problem 2
Integration by Parts:
Substitute
:
Problem 3
Application of this stuff: temperature at a point Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (x,y,z)i}
at time
Newton's law of cooling states that heat flows against the gradient of the temperature and is proportional to the difference in temperature between two objects:
, where
is a property of the material called heat diffusivity.
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle u_{t}=-{\vec {\nabla }}\cdot (-\alpha {\vec {\nabla }}u)=\alpha \Delta u}
For an insulated region
,
on a boundary. There is no gradient of heat leaving (pointing outward) along the boundary.
Show that
Since we are not integrating with respect to
, so we can differentiate the inside:
Substitute
:
Since
does not change over time, temperature enclosed must be constant.
Problem 4
The last step is from problem 3. Continue from there
Problem 6
Kinetic Energy and Potential Energy...
Find
to show that energy is conserved.
This is true because
along ∂R so
as well.
Therefore energy is conserved.
Q.E.D.
Note: 
is often written as

, called the "normal derivative of

".