Respuesta :
Answer with Step-by-step explanation:
We are given that all the given functions have continuous second-order partial derivatives.
[tex]z=f(x,y)[/tex]
Where [tex]x=9rcos\theta,y=9r sin\theta[/tex]
We have to find
A.[tex]\frac{\delta z}{\delta r}[/tex]
We know that
[tex]\frac{\delta z}{\delta r}=\frac{\delta z}{\delta x}\frac{\delta x}{\delta r}+\frac{\delta z}{\delta y}\frac{\delta y}{\delta r}[/tex]
Using this formula
[tex]\frac{\delta z}{\delta r}=9cos\theta \frac{\delta z}{\delta r}+9sin\theta\frac{\delta z}{\delta r}[/tex]
[tex]\frac{\delta z}{\delta r}=\frac{x}{r}\frac{\delta z}{\delta r}+\frac{y}{r}\frac{\delta z}\delta y}[/tex]
B.[tex]\frac{\delta z}{\delta \theta}[/tex]
[tex]\frac{\delta z}{\delta \theta}=\frac{\delta z}{\delta x}\frac{\delta x}{\delta\theta }+\frac{\delta z}{\delta y}\frac{\delta y}{\delta\theta}[/tex]
[tex]\frac{\delta z}{\delta \theta}=-9rsin\theta\frac{\delta z}{\delta x}+9rcost\theta\frac{\delta z}{\delta y}[/tex]
[tex]\frac{\delta z}{\delta \theta}=-y\frac{\delta z}{\delta x}+x\frac{\delta z}{\delta y}[/tex]
C.[tex]\frac{\delta^2 z}{\delta r\delta\theta}[/tex]
[tex]\frac{\delta^2 z}{\delta r\delta\theta}=-9sin\theta\frac{\delta z}{\delta x}-y\frac{\delta^2z}{\delta x^2}(9cos\theta)+9 cos\theta\frac{\delta z}{\delta y}+x\frac{\delta^2z}{\delta y^2}(9sin\theta)[/tex]
[tex]\frac{\delta^2 z}{\delta r\delta\theta}=-9sin\theta\frac{\delta z}{\delta x}-(81rsin\theta cos\theta)\frac{\delta^2z}{dx^2}+9cos\theta\frac{\delta z}{\delta y}+(81r cos\theta sin\theta)\frac{\delta^2z}{\delta y^2}[/tex]
[tex]\frac{\delta^2 z}{\delta r\delta\theta}=-\frac{y}{r}\frac{\delta z}{\delta x}-\frac{xy}{r}\frac{\delta^2}{\delta x^2}+\frac{x}{r}\frac{\delta z}{\delta y}+\frac{xy}{r}\frac{\delta^2z}{\delta y^2}[/tex]
a) The complete expression is [tex]\frac{\partial z}{\partial r} = 9\cdot \frac{\partial z}{\partial x} \cdot \cos \theta + 9\cdot \frac{\partial z}{\partial y} \cdot \sin \theta[/tex].
b) The complete expression is [tex]\frac{\partial z}{\partial \theta} = -9\cdot r\cdot \frac{\partial z}{\partial x} \cdot \sin \theta + 9\cdot r\cdot \frac{\partial z}{\partial y}\cdot \cos \theta[/tex].
c) The complete expression is [tex]\frac{\partial^{2}z}{\partial r\,\partial \theta} = -81\cdot r\cdot \sin \theta\cdot \cos \theta\cdot \frac{\partial ^{2}z}{\partial x^{2}}-9\cdot \sin \theta \cdot \frac{\partial z}{\partial x} + 81\cdot r\cdot \sin \theta \cdot \cos \theta \cdot \frac{\partial^{2}z}{\partial^{2}y}+9\cdot \cos \theta \cdot \frac{\partial z}{\partial y}[/tex].
How to determine partial derivatives
We determine all partial derivatives by applying derivative rules:
a) [tex]\frac{\partial z}{\partial r} = \frac{\partial z}{\partial x}\cdot \frac{\partial x}{\partial r} + \frac{\partial z}{\partial y}\cdot \frac{\partial y}{\partial r}[/tex], where [tex]\frac{\partial x}{\partial r} = 9\cdot \cos \theta[/tex] and [tex]\frac{\partial y}{\partial r} = 9\cdot \sin \theta[/tex].
The complete expression is [tex]\frac{\partial z}{\partial r} = 9\cdot \frac{\partial z}{\partial x} \cdot \cos \theta + 9\cdot \frac{\partial z}{\partial y} \cdot \sin \theta[/tex]. [tex]\blacksquare[/tex]
b) [tex]\frac{\partial z}{\partial \theta } = \frac{\partial z}{\partial x}\cdot \frac{\partial x}{\partial \theta} + \frac{\partial z}{\partial y}\cdot \frac{\partial y}{\partial \theta}[/tex], where [tex]\frac{\partial x}{\partial \theta} = -9\cdot r\cdot \sin \theta[/tex] and [tex]\frac{\partial y}{\partial \theta} = 9\cdot r\cdot \cos \theta[/tex].
The complete expression is [tex]\frac{\partial z}{\partial \theta} = -9\cdot r\cdot \frac{\partial z}{\partial x} \cdot \sin \theta + 9\cdot r\cdot \frac{\partial z}{\partial y}\cdot \cos \theta[/tex]. [tex]\blacksquare[/tex]
c) Now we derive an expression for [tex]\frac{\partial^{2}z}{\partial r \,\partial \theta}[/tex]:
[tex]\frac{\partial^{2} z}{\partial r\,\partial \theta} = \frac{\partial^{2} z}{\partial x^{2}}\cdot \frac{\partial x}{\partial r}\cdot \frac{\partial x}{\partial \theta} +\frac{\partial z}{\partial x}\cdot \frac{\partial x}{\partial r \,\partial \theta} + \frac{\partial^{2} z}{\partial y^{2}}\cdot \frac{\partial y}{\partial r}\cdot \frac{\partial y}{\partial \theta} +\frac{\partial z}{\partial y}\cdot \frac{\partial y}{\partial r \,\partial \theta}[/tex], where [tex]\frac{\partial x}{\partial r \,\partial \theta} = -9\cdot \sin \theta[/tex] and [tex]\frac{\partial y}{\partial r \,\partial \theta} = 9\cdot \cos \theta[/tex].
The complete expression is [tex]\frac{\partial^{2}z}{\partial r\,\partial \theta} = \frac{\partial^{2}z}{\partial x^{2}}\cdot (9\cdot \cos \theta) \cdot (-9\cdot r\cdot \sin \theta) + \frac{\partial z}{\partial x} \cdot (-9\cdot \sin \theta) + \frac{\partial^{2}z}{\partial y^{2}}\cdot (9\cdot \sin \theta) \cdot (9\cdot r\cdot \cos \theta)+ \frac{\partial z}{\partial y}\cdot (9\cdot \cos \theta)[/tex]
[tex]\frac{\partial^{2}z}{\partial r\,\partial \theta} = -81\cdot r\cdot \sin \theta\cdot \cos \theta\cdot \frac{\partial ^{2}z}{\partial x^{2}}-9\cdot \sin \theta \cdot \frac{\partial z}{\partial x} + 81\cdot r\cdot \sin \theta \cdot \cos \theta \cdot \frac{\partial^{2}z}{\partial^{2}y}+9\cdot \cos \theta \cdot \frac{\partial z}{\partial y}[/tex]
The complete expression is [tex]\frac{\partial^{2}z}{\partial r\,\partial \theta} = -81\cdot r\cdot \sin \theta\cdot \cos \theta\cdot \frac{\partial ^{2}z}{\partial x^{2}}-9\cdot \sin \theta \cdot \frac{\partial z}{\partial x} + 81\cdot r\cdot \sin \theta \cdot \cos \theta \cdot \frac{\partial^{2}z}{\partial^{2}y}+9\cdot \cos \theta \cdot \frac{\partial z}{\partial y}[/tex]. [tex]\blacksquare[/tex]
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