Considere a seguinte função contínua:
z(u, v, x)=12 u^{3} v \cos (x)
Portanto, sua diferencial é:
d z=M d u+N d v+F d x
Mostre que (calcule os dois lados das equações e verifique que são iguais):
a) \quad\left(\frac{\partial M}{\partial v}\right)_{u, x}=\left(\frac{\partial N}{\partial u}\right)_{v, x}
b) \quad\left(\frac{\partial M}{\partial x}\right)_{v, u}=\left(\frac{\partial F}{\partial u}\right)_{v, x}

Passo 1

O enunciado dá a função contínua
z(u,v,x)=12\,u^{3}v\cos(x)
e define a diferencial
dz = M\,du + N\,dv + F\,dx
ou seja,
M = \frac{\partial z}{\partial u},\quad N = \frac{\partial z}{\partial v},\quad F = \frac{\partial z}{\partial x}.
O que precisamos é calcular M,N,F e verificar as igualdades de derivadas parciais cruzadas:
a) \bigl(\partial M/\partial v\bigr)_{u,x} = \bigl(\partial N/\partial u\bigr)_{v,x}
b) \bigl(\partial M/\partial x\bigr)_{u,v} = \bigl(\partial F/\partial u\bigr)_{v,x}.

Passo 2

Calculamos as derivadas parciais de z:
M = \frac{\partial z}{\partial u} = \frac{\partial}{\partial u}\bigl(12\,u^{3}v\cos(x)\bigr) = 36\,u^{2}v\cos(x),
N = \frac{\partial z}{\partial v} = \frac{\partial}{\partial v}\bigl(12\,u^{3}v\cos(x)\bigr) = 12\,u^{3}\cos(x),
F = \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}\bigl(12\,u^{3}v\cos(x)\bigr) = -12\,u^{3}v\sin(x).

Passo 3

Calculamos as derivadas mistas para o item (a):
\left(\frac{\partial M}{\partial v}\right)_{u,x} = \frac{\partial}{\partial v}\bigl(36\,u^{2}v\cos(x)\bigr) = 36\,u^{2}\cos(x),
\left(\frac{\partial N}{\partial u}\right)_{v,x} = \frac{\partial}{\partial u}\bigl(12\,u^{3}\cos(x)\bigr) = 36\,u^{2}\cos(x).
Logo,
\left(\frac{\partial M}{\partial v}\right)_{u,x} = \left(\frac{\partial N}{\partial u}\right)_{v,x}.

Passo 4

Calculamos as derivadas mistas para o item (b):
\left(\frac{\partial M}{\partial x}\right)_{u,v} = \frac{\partial}{\partial x}\bigl(36\,u^{2}v\cos(x)\bigr) = -36\,u^{2}v\sin(x),
\left(\frac{\partial F}{\partial u}\right)_{v,x} = \frac{\partial}{\partial u}\bigl(-12\,u^{3}v\sin(x)\bigr) = -36\,u^{2}v\sin(x).
Portanto,
\left(\frac{\partial M}{\partial x}\right)_{u,v} = \left(\frac{\partial F}{\partial u}\right)_{v,x}.

Resposta

a) \displaystyle \bigl(\partial M/\partial v\bigr)_{u,x} = \bigl(\partial N/\partial u\bigr)_{v,x}.
b) \displaystyle \bigl(\partial M/\partial x\bigr)_{u,v} = \bigl(\partial F/\partial u\bigr)_{v,x}.

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