16.1. Definition and Notation

Try these first

1. 1.

   What is a partial derivative?

   Hint

   Think of it as measuring change in one direction.

2. 2.

   Write down the notation for the first-order partial derivative of u with respect to x.

   Hint

   Focus on the variable that is being differentiated.

3. 3.

   What is the primary focus of a partial differential equation?

   • Involves single variables
   • Involves partial derivatives
   • Involves ordinary derivatives

   Hint

   Remember, multiple independent variables are key.

4. 4.

   True or False: Laplace's Equation is a type of linear PDE.

   • True
   • False

   Hint

   Focus on the structure and terms of the equation.

5. 5.

   Prove that if u(x,y) is a solution of Laplace’s Equation in a region, then the average value of u over any circle in that region is equal to the value of u at the center of the circle.

   Hint

   Consider the implications of averaging values.

6. 6.

   Analyze the behavior of a given boundary value problem for Laplace’s Equation in a rectangular domain: Given u(x,0) = f(x) and u(0,y) = g(y), find a method to determine solutions.

   Hint

   Focus on breaking the problem into solvable parts.