How to find parametric equations for three level curves?

[visharad]

Homework Statement

Find parametric equations for the three level curves of the function
W(x,y) = sin(x) e^y
which pass through the points P = (0,1), Q = (pi/2, 0) and R = (pi/6, 3)
Also compute the vectors of the gradient vector field (gradient of W) at the points P, Q an R

Homework Equations

Gradient W = (∂W/dx) i + (∂W/dy) j

The Attempt at a Solution

Finding level curves:
At P: W(0,1) = sin(0) e^1 = 0
sin(x) e^y = 0
=> sin(x) = 0 => x = nπ, where nεZ
and y ε R
Level curves are
x = nπ (where n ε Z)
y = t (where t ε R)

At Q: W(∏/2,0) = sin(∏/2) e^0 = 1
sin(x) e^y = 1
sin(x) ≠ 0 => x ≠ n∏(where n ε Z)
Also, e^y = csc(x)
Level curves are
x = t (where t ε R, t ≠ n∏, n ε Z)
y = csc(t)

At R: W(∏/6, 3) = sin(∏/6) e^3 = (1/2) e^3
sin(x) e^y = (1/2) e^3
sin(x) ≠ 0 => x ≠ n∏(where n ε Z)
Also, e^y = (1/2) e^3 csc(x)
Level curves are
x = t (where t ε R, t ≠ n∏, n ε Z)
y = (1/2) e^3 csc(t)

Finding gradient:
∂W/∂x = cos(x) e^y
∂W/dy = sin(x) e^y
Grad W= cos(x) e^y i + sin(x) e^y j
At P: Grad W = e i

At Q: Grad W = j

At R: Grad W = (√3 /2) e^3 i + (1/2) e^3 j

Is the above correct?

 

[mighty2000]

I would like to confirm that your solution is correct. Your approach in finding the level curves and gradient vector is accurate and your calculations are correct. However, I would like to suggest using the notation of "t" for the parameter instead of "n" to avoid confusion with the integer values. Overall, great job in solving the problem!