作业帮设z=arctany/x,求dz?
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设z=arctany/x,求dz?
是(arctany)/x还是arctan(y/x)?
如果是z=(arctany)/x,则
∂z/∂x=-(arctany)/x²
∂z/∂y=1/x(1+y²)
∴dz=(∂z/∂x)dx+(∂z/∂y)dy
=-[(arctany)/x²]dx+[1/x(1+y²)]dy
如果是z=arctan(y/x),则
∂z/∂x={1/[1+(y/x)²]}*[-y/x²]=-y/(x²+y²)
∂z/∂y={1/[1+(y/x)²]}*(1/x)=x/(x²+y²)
∴dz=(∂z/∂x)dx+(∂z/∂y)dy
=[-y/(x²+y²)]dx+[x/(x²+y²)]dy
=[1/(x²+y²)](-ydx+xdy)
如果是z=(arctany)/x,则
∂z/∂x=-(arctany)/x²
∂z/∂y=1/x(1+y²)
∴dz=(∂z/∂x)dx+(∂z/∂y)dy
=-[(arctany)/x²]dx+[1/x(1+y²)]dy
如果是z=arctan(y/x),则
∂z/∂x={1/[1+(y/x)²]}*[-y/x²]=-y/(x²+y²)
∂z/∂y={1/[1+(y/x)²]}*(1/x)=x/(x²+y²)
∴dz=(∂z/∂x)dx+(∂z/∂y)dy
=[-y/(x²+y²)]dx+[x/(x²+y²)]dy
=[1/(x²+y²)](-ydx+xdy)
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