91影视

We will compute the partial derivatives of f1, f2, and f3 with respect to variables x, y, and z. We will denote partial derivatives as df/dx, df/dy, and df/dz. For f1(x, y, z): - Partial derivative with respect to x: \(\frac{\partial f_1}{\partial x}=ye^{xy}\) - Partial derivative with respect to y: \(\frac{\partial f_1}{\partial y}=xe^{xy}\) - Partial derivative with respect to z: \(\frac{\partial f_1}{\partial z}=-2\cos z\sin z\) For f2(x, y, z): - Partial derivative with respect to x: \(\frac{\partial f_2}{\partial x}=yz\) - Partial derivative with respect to y: \(\frac{\partial f_2}{\partial y}=xz\) - Partial derivative with respect to z: \(\frac{\partial f_2}{\partial z}=xy+e^{-z}\) For f3(x, y, z): - Partial derivative with respect to x: \(\frac{\partial f_3}{\partial x}=z\cosh(xz)\) - Partial derivative with respect to y: \(\frac{\partial f_3}{\partial y}=2y\) - Partial derivative with respect to z: \(\frac{\partial f_3}{\partial z}=x\cosh(xz)\)

We will compute the partial derivatives of g1, g2, and g3 with respect to variables x1, x2, x3, and x4. For g1(x1, x2, x3, x4): - Partial derivative with respect to x1: \(\frac{\partial g_1}{\partial x_1}=\frac{x_1}{\sqrt{x_1^2+x_2^2+1}}\) - Partial derivative with respect to x2: \(\frac{\partial g_1}{\partial x_2}=\frac{x_2}{\sqrt{x_1^2+x_2^2+1}}\) - Partial derivative with respect to x3: \(\frac{\partial g_1}{\partial x_3}=0\) - Partial derivative with respect to x4: \(\frac{\partial g_1}{\partial x_4}=-1\) For g2(x1, x2, x3, x4): - Partial derivative with respect to x1: \(\frac{\partial g_2}{\partial x_1}=-x_3^2\sin(x_1x_3^2)\) - Partial derivative with respect to x2: \(\frac{\partial g_2}{\partial x_2}=0\) - Partial derivative with respect to x3: \(\frac{\partial g_2}{\partial x_3}=-2x_1x_3\sin(x_1x_3^2)\) - Partial derivative with respect to x4: \(\frac{\partial g_2}{\partial x_4}=e^{x_4}\) For g3(x1, x2, x3, x4): - Partial derivative with respect to x1: \(\frac{\partial g_3}{\partial x_1}=0\) - Partial derivative with respect to x2: \(\frac{\partial g_3}{\partial x_2}=x_3\) - Partial derivative with respect to x3: \(\frac{\partial g_3}{\partial x_3}=x_2\) - Partial derivative with respect to x4: \(\frac{\partial g_3}{\partial x_4}=\frac{2x_4}{1+x_4^2}\)

We will now arrange the partial derivatives obtained above into their respective Jacobi matrices. For f, arrange partial derivatives like this: $$ J_f = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} & \frac{\partial f_1}{\partial z} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} & \frac{\partial f_2}{\partial z} \\ \frac{\partial f_3}{\partial x} & \frac{\partial f_3}{\partial y} & \frac{\partial f_3}{\partial z} \end{bmatrix} = \begin{bmatrix} ye^{xy} & xe^{xy} & -2\cos z\sin z \\ yz & xz & xy+e^{-z} \\ z\cosh(xz) & 2y & x\cosh(xz) \end{bmatrix} $$ For g, arrange the partial derivatives like this: $$ J_g = \begin{bmatrix} \frac{\partial g_1}{\partial x_1} & \frac{\partial g_1}{\partial x_2} & \frac{\partial g_1}{\partial x_3} & \frac{\partial g_1}{\partial x_4} \\ \frac{\partial g_2}{\partial x_1} & \frac{\partial g_2}{\partial x_2} & \frac{\partial g_2}{\partial x_3} & \frac{\partial g_2}{\partial x_4} \\ \frac{\partial g_3}{\partial x_1} & \frac{\partial g_3}{\partial x_2} & \frac{\partial g_3}{\partial x_3} & \frac{\partial g_3}{\partial x_4} \end{bmatrix} = \begin{bmatrix} \frac{x_1}{\sqrt{x_1^2+x_2^2+1}} & \frac{x_2}{\sqrt{x_1^2+x_2^2+1}} & 0 & -1 \\ -x_3^2\sin(x_1x_3^2) & 0 & -2x_1x_3\sin(x_1x_3^2) & e^{x_4} \\ 0 & x_3 & x_2 & \frac{2x_4}{1+x_4^2} \end{bmatrix} $$ The Jacobi matrices, Jf and Jg, have now been computed.