On $R^3$ consider the vector field $X= x\frac{\partial}{\partial x}+ y\frac{\partial}{\partial y} $and the function $f(x, y, z) = x^2+y^2+z^2$.
Calculate the directional derivative $Xf$.
I know how to do the exercise if you have to solve it with a vector or based on a point but not having this information I don't know how to do it.
It's simple, you only have to solve in this way: $$Xf=Xf(x,y,z)=(x\frac{\partial}{\partial x}+ y\frac{\partial}{\partial y})(x^{2}+y^{2}+z^{2})$$ From this you find: $$...=x\frac{\partial x^{2}}{\partial x} + y\frac{\partial y^{2}}{\partial y} =2x^{2}+2y^{2}$$
ll other terms are equal to zero.
I hope it will help you.