Resolución ejercicio 7 pagina 153

Sean f yg : R3 ---> R

demostrar que

grad(fg) = f·grad(g) + g·grad(f)

Los puntos son producto de un escalar por un vector, no el producto escalar de dos vectores.

$$\begin{align}&\nabla f=\frac{\partial f}{\partial x}i+\frac{\partial f}{\partial y}j+\frac{\partial f}{\partial z}k\\ &\\ &\\ &\\ &\nabla g=\frac{\partial g}{\partial x}i+\frac{\partial g}{\partial y}j+\frac{\partial g}{\partial z}k\\ &\\ &\\ &\nabla (fg)=\frac{\partial fg}{\partial x}i+\frac{\partial fg}{\partial y}j+\frac{\partial fg}{\partial z}k=\\ &\\ &\left(\frac{\partial f}{\partial x}g+f \frac{\partial g}{\partial x}\right)i+\left(\frac{\partial f}{\partial y}g+f \frac{\partial g}{\partial y}\right)j+\left(\frac{\partial f}{\partial z}g+f \frac{\partial g}{\partial z}\right)i=\\ &\\ &\\ &\\ &f\left(\frac{\partial g}{\partial x}i+\frac{\partial g}{\partial y}j+\frac{\partial g}{\partial z}k\right)+g\left(\frac{\partial f}{\partial x}i+\frac{\partial f}{\partial y}j+\frac{\partial f}{\partial z}k\right)=\\ &\\ &\\ &\\ &f\,\nabla g+g\,\nabla f\end{align}$$

Y eso es todo.