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.
it this correct for 2. a)
f is differentiable at (a,b) iff there exists a linear mapping L such that lim((f(a+h,b+k)-f(a,b)-L(h,k))/|(h,k)|,(h,k)->(0,0))=0. ∂f/∂x(a,b)=lim((f(a+h,b)-f(a,b))/h,h->0), and ∂f/∂y(a,b)=lim((f(a,b+k)-f(a,b))/k,k->0).
does L=grad f; that is, L(h,k)=(∂f/∂x,∂f/∂y).(h,k)=h*∂f/∂x+k*∂f/∂y.
what do i do next?
b) Prove from the definition in a) that the function f defined by f(x,y) =xy(x+y) is differentiable at every point of it’s domain
c) If g(x, y) = xy prove that g is not differentiable at (0, b) for any non- zero value of b
d) Prove that the function g of part c) is differentiable at all points (a,b) for which a is not zero and at the origin (0,0)
Am mainly stuck on parts c) and d) but am unsure if am an starting along the right vein with the first parts
(a, b) and prove that if f is differentiable at (a, b) then both these partial derivatives exist.
it this correct for 2. a)
f is differentiable at (a,b) iff there exists a linear mapping L such that lim((f(a+h,b+k)-f(a,b)-L(h,k))/|(h,k)|,(h,k)->(0,0))=0. ∂f/∂x(a,b)=lim((f(a+h,b)-f(a,b))/h,h->0), and ∂f/∂y(a,b)=lim((f(a,b+k)-f(a,b))/k,k->0).
does L=grad f; that is, L(h,k)=(∂f/∂x,∂f/∂y).(h,k)=h*∂f/∂x+k*∂f/∂y.
what do i do next?
b) Prove from the definition in a) that the function f defined by f(x,y) =xy(x+y) is differentiable at every point of it’s domain
c) If g(x, y) = xy prove that g is not differentiable at (0, b) for any non- zero value of b
d) Prove that the function g of part c) is differentiable at all points (a,b) for which a is not zero and at the origin (0,0)
Am mainly stuck on parts c) and d) but am unsure if am an starting along the right vein with the first parts