1) Introduction and Motivation

Question 1

How are ordinary differentiation and partial differentiation of functions denoted and distinguished

Answer 1

• Ordinary differentiation - du/dx
• Partial differentiation - ∂u/∂x and ∂u/∂y

Question 2

What is the Chain Rule for Functions of Multiple Variables

Answer 2

Question 3

What is the Chain Rule for Coordinate Transformations

Answer 3

Question 4

What is the Taylor Series expansion for a function around the point a

Answer 4

Question 5

What is the Wave Equation

Answer 5

Question 6

What is the unstead Heat Equation

Answer 6

Question 7

What is Laplace’s Equation

Answer 7

Question 8

Summarise the derivation of the damped wave equation

Answer 8

Question 9

Summarise the derivation of the unstead heat equation

Answer 9

Question 10

What are the limitations of using a one-dimensional heat model for a heated spoon

Answer 10

• Model assumes insulation at the top and bottom of the spoon, simplifying the actual conditions
• The spoon is surrounded by air at a constant room temperature, making it a three-dimensional heat transfer problem
• Model Justification: This model can still provide a reasonable approximation of the temperature distribution near the middle of the spoon