I am given Z = f (x, y), where x= r cosθ and y=r sinθ
I found
∂z/∂r = ∂z/∂x ∂x/∂r + ∂z/∂y ∂y/∂r = (cos θ) ∂z/∂x + (sin θ) ∂z/∂y and
∂z/∂θ = ∂z/∂x ∂x/∂θ + ∂z/∂y ∂y/∂θ= (-r sin θ) ∂z/∂x + (r cos θ) ∂z/∂y
I need to show that
∂z/∂x = cos θ ∂z/∂r - 1/r * sin θ ∂z/∂θ and
∂z/∂y = sin...
Given the first order ODE y' = 2 * y^3 * x, plot the isoclines corresponding to slopes m = +1 and m = -1.
My answer:
The isoclines are given by m= 2 * y^3 * x or y=(m/(2*x))^1/3
The slope elements on the isocline y=+- (m/(2*x))^1/3 all have gradient m
For m = 1, the isocline is y =...
Homework Statement
Applying the definition of a limit to show that
lim ((x^3 * y(y-1) ) / (x^2 + (y-1)^2) = 0 as (x,y) approaches (0,1)
The Attempt at a Solution
|x| = sqrt(x^2) <= sqrt((x^2 + (y-1)^2))
|y-1|=sqrt((y-1)^2)<= sqrt((x^2 + (y-1)^2))
|y|<= |y-1| + 1 via the...