4. Compute ∂z/∂r +
∂z/∂s if z = sin u cos v, u = (r + s)2, and v = (r − s)2.
To compute ∂z/∂r + ∂z/∂s, we need to find the partial derivatives of z with respect to both r and s and then add them together.
Let's start by finding ∂z/∂r:
Step 1: Compute ∂z/∂u:
To find ∂z/∂u, we need to differentiate z with respect to u while treating v as a constant.
z = sin u cos v
∂z/∂u = cos u cos v
Step 2: Compute ∂u/∂r:
To find ∂u/∂r, we differentiate u with respect to r while treating s as a constant.
u = (r + s)^2
∂u/∂r = 2(r + s)
Step 3: Find ∂z/∂r:
Using the chain rule, we can compute ∂z/∂r by multiplying ∂z/∂u and ∂u/∂r.
∂z/∂r = ∂z/∂u * ∂u/∂r
= cos u cos v * 2(r + s)
= 2(r + s) cos u cos v
Now let's find ∂z/∂s:
Step 1: Compute ∂z/∂u:
We already computed ∂z/∂u in the previous step.
∂z/∂u = cos u cos v
Step 2: Compute ∂u/∂s:
To find ∂u/∂s, we differentiate u with respect to s while treating r as a constant.
u = (r + s)^2
∂u/∂s = 2(r + s)
Step 3: Find ∂z/∂s:
Again, using the chain rule, we can compute ∂z/∂s by multiplying ∂z/∂u and ∂u/∂s.
∂z/∂s = ∂z/∂u * ∂u/∂s
= cos u cos v * 2(r + s)
= 2(r + s) cos u cos v
Finally, we can compute ∂z/∂r + ∂z/∂s by adding ∂z/∂r and ∂z/∂s together:
∂z/∂r + ∂z/∂s = (2(r + s) cos u cos v) + (2(r + s) cos u cos v)
= 4(r + s) cos u cos v
Therefore, ∂z/∂r + ∂z/∂s = 4(r + s) cos u cos v.