散度计算器

计算 $$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle$$$。

按定义，$$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = \nabla\cdot \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle$$$，或者等价地，$$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle\cdot \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle$$$，其中 $$$\cdot$$$ 是 点积运算符。

因此，$$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = \frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) + \frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) + \frac{\partial}{\partial z} \left(e^{z}\right)$$$。

求分量1关于$$$x$$$的偏导数：$$$\frac{\partial}{\partial x} \left(\sin{\left(x y \right)}\right) = y \cos{\left(x y \right)}$$$（步骤详见导数计算器）。

求分量2关于$$$y$$$的偏导数：$$$\frac{\partial}{\partial y} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$（步骤详见导数计算器）。

求分量3关于$$$z$$$的偏导数：$$$\frac{\partial}{\partial z} \left(e^{z}\right) = e^{z}$$$（步骤详见导数计算器）。

现在，只需将上述表达式相加即可得到散度：$$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = - x \sin{\left(x y \right)} + y \cos{\left(x y \right)} + e^{z}$$$

$$$\operatorname{div} \left\langle \sin{\left(x y \right)}, \cos{\left(x y \right)}, e^{z}\right\rangle = - x \sin{\left(x y \right)} + y \cos{\left(x y \right)} + e^{z}$$$A