Задача 57937 Найти dx/dt, если \( z = z(x;y) \), \( x...

5f91af2c80c7303e80e6594e

16.02.2021 20:36:26

Найти dx/dt, если \( z = z(x;y) \), \( x = x(t) \), \( y = y(t) \):

11.4.4. \( z = x^2 + y^2 + xy \), \( x = a \sin t \), \( y = a \cos t \).

5f3ea7e3faf909182968ddd9

16.02.2021 22:00:36

★

[m]\frac{ dz }{ dt }=\frac{ ∂ z }{ ∂ x }\cdot \frac{ d x }{ d t }+\frac{ ∂ z }{ ∂ y }\cdot \frac{ d y }{ d t }[/m]



[m]\frac{ ∂ z }{ ∂ x }=(x^2+y^2+xy)`_{x}=2x+y[/m]

[m]\frac{ ∂ z }{ ∂ y }=(x^2+y^2+xy)`_{y}=2y+x[/m]


[m]\frac{ d x }{ d t }=(asint)`=acost[/m]
[m]\frac{ d y }{ d t }=(acost)`=-asint[/m]


[m]\frac{ dz }{ dt }=(2x+y)\cdot (acost)+(2y+x)\cdot (-asint)=a\cdot( (2x+y)cost-(2y+x)sint)[/m]