Why does the gradient vector point straight outward from a graph?

[gikiian]

A gradient vector points out of a graph (or a surface in 3D case). Locally, it makes an angle of 90 degrees with the graph at a particular point. Why is that so?

Thanks.

 

[kdbnlin78]

Hi.

The gradient vector measures the change and direction of a scalar field. The direction of the gradient is expressed in terms of unit vectors (in 3-dimensions, say) and points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

 

[HallsofIvy]

Another way of looking at it is that the "directional derivative", the rate of change of function f(x,y,z) as you move in the direction of unit vector $\vec{v}$, is given by $\nabla f\cdot\vec{v}$. If the function is given implicitely by f(x,y,z)= 0 (or any constant, then on the surface f is a constant and so it derivative is 0 in any direction tangent to surface: the dot product of $\nabla f\cdot \vec{v}$, with $\vec{v}$ tangent to the surface, is 0 so $\nabla f$ is perpendicular to the surface.