1) $$ \begin{array}{l} ext { Find } \frac{\partial w}{\partial s} \frac{\delta \partial w}{\partial t} ext { when } s=1 ext { and } \ t=2 \pi ext { for } w=x y+y z+x z \ ext { where } x=s \cos t, y=s \sin t, z=t .\end{array} $$

Question

1) $$ \begin{array}{l}	ext { Find } \frac{\partial w}{\partial s} \frac{\delta \partial w}{\partial t} 	ext { when } s=1 	ext { and } \ t=2 \pi 	ext { for } w=x y+y z+x z \ 	ext { where } x=s \cos t, y=s \sin t, z=t .\end{array} $$

UpStudy AI · Accepted Answer

To solve the problem, we'll follow these steps:

1. **Express $$ w $$ in terms of $$ s $$ and $$ t $$:**
   $$
   w = xy + yz + xz
   $$
   Given:
   $$
   x = s \cos t, \quad y = s \sin t, \quad z = t
   $$
   Substitute these into the expression for $$ w $$:
   $$
   w = (s \cos t)(s \sin t) + (s \sin t)(t) + (s \cos t)(t) = s^2 \cos t \sin t + s t \sin t + s t \cos t
   $$

2. **Compute the partial derivatives $$ \frac{\partial w}{\partial s} $$ and $$ \frac{\partial w}{\partial t} $$:**

- **Partial derivative with respect to $$ s $$:**
     $$
     \frac{\partial w}{\partial s} = 2s \cos t \sin t + t \sin t + t \cos t
     $$
   
   - **Partial derivative with respect to $$ t $$:**
     $$
     \frac{\partial w}{\partial t} = s^2 (\cos^2 t - \sin^2 t) + s \sin t + s t \cos t + s \cos t - s t \sin t
     $$
     Simplifying:
     $$
     \frac{\partial w}{\partial t} = s^2 \cos(2t) + s \sin t + s \cos t + s t \cos t - s t \sin t
     $$

3. **Evaluate the partial derivatives at $$ s = 1 $$ and $$ t = 2\pi $$:**

- **Evaluate $$ \frac{\partial w}{\partial s} $$ at $$ s = 1 $$, $$ t = 2\pi $$:**
     $$
     \frac{\partial w}{\partial s} = 2(1) \cos(2\pi) \sin(2\pi) + (2\pi) \sin(2\pi) + (2\pi) \cos(2\pi)
     $$
     $$
     = 2 \cdot 1 \cdot 0 + 2\pi \cdot 0 + 2\pi \cdot 1 = 2\pi
     $$
   
   - **Evaluate $$ \frac{\partial w}{\partial t} $$ at $$ s = 1 $$, $$ t = 2\pi $$:**
     $$
     \frac{\partial w}{\partial t} = 1^2 \cos(4\pi) + 1 \sin(2\pi) + 1 \cos(2\pi) + 1 \cdot 2\pi \cos(2\pi) - 1 \cdot 2\pi \sin(2\pi)
     $$
     $$
     = 1 \cdot 1 + 0 + 1 \cdot 1 + 2\pi \cdot 1 - 0 = 2 + 2\pi
     $$

4. **Multiply the evaluated partial derivatives:**
   $$
   \frac{\partial w}{\partial s} \cdot \frac{\partial w}{\partial t} = 2\pi \cdot (2 + 2\pi) = 4\pi (1 + \pi)
   $$

**Final Answer:**
$$
\frac{\partial w}{\partial s} \cdot \frac{\partial w}{\partial t} = 4\pi (1 + \pi)
$$
$$ \frac{\partial w}{\partial s} \cdot \frac{\partial w}{\partial t} = 4\pi (1 + \pi) $$

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