(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

Gradient W = (∂W/dx) i + (∂W/dy) j

3. 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?

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# Homework Help: How to find parametric equations for three level curves?

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