Usando a regra da cadeia para z=f(x, y) e w=f(x, y, z), calcule:
partial zpartial t, partial zpartial s e partial wpartial t, partial wpartial s e partial wpartial r
w=x^2+y^2+z^2, x=r cos (s), y=r sen(t) sen(s), z=r cos (t)

Passo 1

O enunciado envolve duas composições:
z é dada genericamente por
z = f(x,y),
com
x = rcos(s), y = rsin(t)sin(s),
e em seguida
w = x^2 + y^2 + z^2,
onde agora z = rcos(t).
Queremos calcular
partial zpartial t,
partial zpartial s
e
partial wpartial t,
partial wpartial s,
partial wpartial r
usando a regra da cadeia.

Passo 2

Aplicando a regra da cadeia a z=f(x,y), temos
partial zpartial t = fx,partial xpartial t + fy,partial ypartial t
Como
partial xpartial t=0, partial ypartial t = rcos(t),sin(s),
segue
partial zpartial t = fy;rcos(t)sin(s)

Passo 3

De modo semelhante,
partial zpartial s = fx,partial xpartial s + fy,partial ypartial s,
onde
partial xpartial s=-,rsin(s), partial ypartial s=rsin(t)cos(s)
Logo
partial zpartial s = -,fx,rsin(s) + fy,rsin(t)cos(s)

Passo 4

Para
w=x^2+y^2+z^2
temos
wx=2x, wy=2y, wz=2z
E das parametrizações:
&x=rcos(s), &&y=rsin(t)sin(s), &&z=rcos(t), &xt=0, &&yt=rcos(t)sin(s), &&zt=-rsin(t), &xs=-rsin(s), &&ys=rsin(t)cos(s), &&zs=0, &xr=cos(s), &&yr=sin(t)sin(s), &&zr=cos(t)

Passo 5

Agora
partial wpartial t = wx,xt + wy,yt + wz,zt =2xcdot0 + 2y,(rcos(t)sin(s)) + 2z,(-rsin(t))
Substituindo x=rcos(s), y=rsin(t)sin(s), z=rcos(t), obtém-se
partial wpartial t =2r^2sin(t)cos(t)sin^2(s) -2r^2sin(t)cos(t) =2r^2sin(t)cos(t)l(sin^2(s)-1r) =-2r^2sin(t)cos(t)cos^2(s)

Passo 6

Analogamente,
partial wpartial s =2x,(-rsin(s)) + 2y,(rsin(t)cos(s)) +2zcdot0
=-2r^2cos(s)sin(s) +2r^2sin^2(t)sin(s)cos(s) =2r^2sin(s)cos(s)l(sin^2(t)-1r) =-2r^2sin(s)cos(s)cos^2(t)

Passo 7

Por fim,
partial wpartial r =2x,cos(s) + 2y,sin(t)sin(s) + 2z,cos(t)
=2rcos^2(s)+2rsin^2(t)sin^2(s)+2rcos^2(t) =2rl[cos^2(s)+sin^2(t)sin^2(s)+cos^2(t)r]

Resposta

partial zpartial t = fy,rcos(t)sin(s), partial zpartial s = -,fx,rsin(s)+fy,rsin(t)cos(s),
partial wpartial t =-2r^2sin(t)cos(t)cos^2(s), partial wpartial s =-2r^2sin(s)cos(s)cos^2(t), partial wpartial r =2rl[cos^2(s)+sin^2(t)sin^2(s)+cos^2(t)r]

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