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

  1. Jul 21, 2011 #1
    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.
     
  2. jcsd
  3. Jul 21, 2011 #2
    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.
     
  4. Jul 21, 2011 #3

    HallsofIvy

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    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 [itex]\vec{v}[/itex], is given by [itex]\nabla f\cdot\vec{v}[/itex]. 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 [itex]\nabla f\cdot \vec{v}[/itex], with [itex]\vec{v}[/itex] tangent to the surface, is 0 so [itex]\nabla f[/itex] is perpendicular to the surface.
     
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