Gradient vector over an area of a surface

In summary, the conversation discusses the concept of the gradient vector on a 3d surface and its relation to the maximum and minimum rates of change. The idea of a tilted plane with a divet that does not follow this rule is brought up and it is concluded that this would not be a continuous function at the point in question. The question then arises of how to compute gradients over an arbitrary area on a surface and if this can be extended to a curve running along the surface. The conversation also touches on the concept of vector fields and differential equations in relation to the gradient.
  • #1
trancefishy
75
0
what follows is a question I asked myself, the answer I figured out, and the new question that arose as a result.

I was thinking about the gradient vector on a 3d surface, and how it shows the direction of the max rate of change at a point. the 2 directions perpendicular to it are tangent to the level curve thus represent zero change, and the direction opposite the gradient represents the minimum rate of change.

so i was trying to think of a surface where this was not true. say, a tilted plane, where, halfway between the origninal min r.o.c. and the zero r.o.c. direction, there was a divet than ran "down". this would then be the min r.o.c. and certainly not perpendicular to the zero r.o.c. direction.

I concluded that this surface would not be a continuous function at the point in question, due to the divet needing to stop abruptly. i had to remind myself that i was talking about a point, not a very small area.

so, how would one compute these gradients over an arbitrary area on a surface? would this be extendable to a curve that ran along the surface?
 
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  • #2
You say "the gradient vector on a 3d surface" but I think you mean the gradient of the function defining the surface. I.e.- f(x,y,z)= constant so you can think of the surface as a "level surface" of f(x,y,z). If that is what you mean then grad f is always normal to the surface. Further, its length, times dA, is the "differential of area" and so grad f itself can be thought of as the "vector differential of area".
 
  • #3
nope, not really what i was going at at all.

state differently, perhaps more simply, is I want to know how to find the equivalent of a gradient vector of a surface f(x,y) over an arbitrary area, not just a specific point.

now that i think of it a little more, this seems a bit absurd. an analogy is, say i have a garden that's a bit uneven and it's raining. normally the opposite direction of the gradient(at the point I'm standing) woudl tell me which direction the water would flow away from my feet. what i want to know, is taking the entire area of the garden into account, what are all the different rates of flow, max, min, etc for the whole area. we are assuming that the function defining the surface of my garden is continuous and differentiable. etc.

this is beginning to sound like a vector field now...
 
  • #4
trancefishy said:
nope, not really what i was going at at all.

state differently, perhaps more simply, is I want to know how to find the equivalent of a gradient vector of a surface f(x,y) over an arbitrary area, not just a specific point.

now that i think of it a little more, this seems a bit absurd. an analogy is, say i have a garden that's a bit uneven and it's raining. normally the opposite direction of the gradient(at the point I'm standing) woudl tell me which direction the water would flow away from my feet. what i want to know, is taking the entire area of the garden into account, what are all the different rates of flow, max, min, etc for the whole area. we are assuming that the function defining the surface of my garden is continuous and differentiable. etc.

this is beginning to sound like a vector field now...

That's exactly what it sounds like! And, therefore, equivalent to a differential equation where the gradient itself is a derivative.
 

What is a gradient vector over an area of a surface?

A gradient vector over an area of a surface is a mathematical concept that represents the direction and magnitude of the steepest increase of a scalar field on a given surface. It is commonly used in fields such as physics and engineering to analyze the behavior of various physical quantities over a surface.

How is a gradient vector over an area of a surface calculated?

A gradient vector over an area of a surface is calculated by taking the partial derivatives of the scalar field with respect to the coordinates of the surface. These partial derivatives are then combined to form a vector that points in the direction of the steepest increase of the scalar field.

What is the significance of a gradient vector over an area of a surface?

The gradient vector over an area of a surface provides important information about the behavior of a scalar field on that surface. It can help determine the direction of maximum change of the scalar field and can be used to optimize various physical processes on the surface.

Can a gradient vector over an area of a surface be negative?

Yes, a gradient vector over an area of a surface can be negative. This indicates a decrease in the scalar field in the direction of the vector, as opposed to an increase. The magnitude of the vector represents the rate of change of the scalar field, so a negative gradient vector indicates a decrease in the scalar field at a faster rate.

How is a gradient vector over an area of a surface used in real-world applications?

The concept of a gradient vector over an area of a surface has many real-world applications, such as in fluid dynamics, heat transfer, and optimization problems. It is also used in computer graphics and image processing to create realistic lighting and shading effects on surfaces.

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