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Question
Let U(x, y) = ex sin y where x = st2, y = s2t, s, t ∈ R. Find `(del"U")/(del"s"), (del"u")/(del"t")` and evaluate them at s = t = 1
Solution
U(x, y) = ex sin y where x = st2, y = s2t
`(del"U")/(del"s") = (del"U")/(delx) (delx)/(del"s") + (del"U")/(dely) (dely)/(del"s")`
`(del"U")/(delx) = "e"^x siny, (delx)/(del"t") = 2"st", (delx)/(del"s") = "t"^2`
`(del"U")/(dely) = "e"^x cosy, (dely)/(del"t") = "s"^2, (dely)/(del"s") = 2"st"`
`(del"U")/(del"s") = "e"^x siny "t"^2 + "e"^x cosy (2"st")`
= est2 sin (s2t) t2 + est2 cos(s2t) 2st
= est2 [t2 sin (s2t) + 2st cos (s2t)]
= t ex [t sin(s2t) + 2s cos (s2t)]
`(del"U")/(del"t")` = ex sin y 2st + ex cos y (s2)
= est2 sin(s2t) 2st + est2 cos(s2t) s2
= s est2 [2t sin (s2t) + s cos(s2t)]
At s = t = 1
`(del"U")/(del"s")` e[sin(1) + 2 cos (1)]
= e[sin(1) + 2 cos (1)]
`(del"U")/(del"t")` = e[2 sin(1) + cos (1)]
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