limits, continuity and Cauchy-Riemann equations

1. a) If f is a real-valued function defined throughout R², except at the origin (0, 0), define what is meant by the statement: f (x, y)→ L as (x, y)→ (0, 0).

2. a) Define what it means to describe a function f of two real variables as differentiable at (a, b)? Define (as limits) the partial derivatives df/dx and df/dy at (a, b) and prove that if f is differentiable at (a, b) then both these partial derivatives exist

b) If g(x, y) = xy prove that g is not differentiable at (0, b) for any non-value of b

1. a) The Cauchy-Riemann equation is the name given to the following pair of equations,

∂u/∂x=∂v/∂y and ∂u/∂y= -∂v/∂x which connects the partial derivatives of two functions u(x, y) and v(x, y)

i) if u(x, y) =e^x cosy and v(x, y) =e^x siny, how do I prove that these functions satisfy the Cauchy-Riemann equations

ii) if u(x, y) =½ In(x² +y²) and v(x, y) = tanֿ¹(y/x), how do I prove that theses functions satisfy the Cauchy-Riemann equations

iii) if u and v are any functions that satisfy the Cauchy-Riemann equations,

how do I prove that ∂²u/∂x² + ∂²u/∂y²=0

b) If f is a real valued function of two variables, the set of points (x, y) for which f(x, y)=c, for some value of the constant c, is called a level curve( or contour line) of the function. How do I illustrate the level curves for the following functions:

i) f(x, y) = x² +y²

ii) g(x, y) = xy

How would I calculate the gradient vectors of these functions and confirm in each case that the direction of this vector at any point is normal to the level curve passing through it