IMPORTANT - what is the geometric intepretation of the gradient vector?

[richardlhp]

IMPORTANT! ---- what is the geometric intepretation of the gradient vector?

Assume the situation in which I have a slope, a component of a function dependent on x and y, which is at an angle to the xy plane. The gradient vector would be perpendicular to the tangent plane at the point in which i want to find the gradient vector.

However, if i plot the function in the form of a topo map with contour lines, the gradient vector will be perpendicular to the level curve directly towards the higher values of the function parallel to the xy plane.

Hence, we see a contradiction. What i know may be wrong, but can sb clarify this with me and give me an intuitive explanation on the geographical interpretation of the gradient? I know all the math, but I need to UNDERSTAND!

 

[HallsofIvy]

I'm sorry but I see no contradiction here! You have a three dimensional situation and say that the gradient vector is perpendicular to surfaces of constant value. Then, in a two dimensional situation, the gradient vector is perpendicular to curves of constant value.

Looks to me like those are saying the same thing- not a contradiction.

 

[nonequilibrium]

The gradient vector in the 2nd case is not the same gradient vector: its z-component is left out, and thus lies in the xy-plane. x and y-components are identical.

 

[richardlhp]

Thanks for your help. So can i say that the gradient vector shown on the contour plot is just a projection of the actual vector in euclidean space to the xy plane?

 

[richardlhp]

[URL]http://www2.seminolestate.edu/lvosbury/images/Sect127No10Pic2.gif[/URL]

Can someone help me visualise and tell me how the gradient vector is going to look like if I plot this function on a contour plot?

 

[chemicalsss]

[URL]http://www2.seminolestate.edu/lvosbury/images/Sect127No10Pic2.gif[/URL]

Can someone help me visualise and tell me how the gradient vector is going to look like if I plot this function on a contour plot?

That surface in the plot is not the plot of the function, but a plot of the level surface. The function is somewhere else, increasing in the direction of the red arrow. (It might be easier to visualise for a 2-variable than a 3-variable plot)

 

[StalkerM]

Hi, the gradient vector show how (and how much) the function is changing. It points in the direction of maximum change. Actually if you want to know how the function behaves in a specific direction (indicated by a unit vector pointing at that direction) you are interested in the component along that direction.

 

[StalkerM]

In a contour plot the gradient vector is always perpendicualar to the level surface (i.e. surface with costant value ). In fact the component along the costant-surface MUST be zero by definition.

 

[LukeD]

The gradient vector is exactly as you say: it is a vector that is perpendicular to the level surfaces. The length of the gradient vector is proportional to the density of level surfaces. Drawing a gradient vector requires you both knowing the level surfaces and having a ruler to do some measurements.

There is also the gradient co-vector, which is slightly different. Rather than being an arrow that's perpendicular to the level surfaces near a point, it actually is the level surfaces near the point. This is actually a much more useful object in Calculus, but the gradient vector has its uses too.