# What does "space" refer to in this sentence (from wikipedia article on gradients)?

• Jun 11th 2013, 04:45 PM
Elusive1324
What does "space" refer to in this sentence (from wikipedia article on gradients)?
"In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase. In simple terms, the variation in space of any quantity can be represented (e.g. graphically) by a slope. The gradient represents the steepness and direction of that slope."

Gradient - Wikipedia, the free encyclopedia

I've heard of variations (i.e., change in the dependent quantity) but I'm not sure what "space" refers to. I've taken linear algebra and multi calc. so I'm familiar with the notion of a field. Does space refer to the target space of a function?
• Jun 11th 2013, 06:13 PM
hollywood
Re: What does "space" refer to in this sentence (from wikipedia article on gradients)
You're probably thinking about it too much - it's not a technical term. It just means how some quantity changes based on position - for example, the temperature changes as you move north/south/east/west/up/down. The temperature is also changing with time, which is an example of what it *doesn't* mean.

The gradient of a real-valued function F(x,y,z) is given by $\frac{\partial{F}}{\partial{x}}\,\bold{i} + \frac{\partial{F}}{\partial{y}}\,\bold{j} + \frac{\partial{F}}{\partial{z}}\,\bold{k}$ and is a vector in the direction of greatest increase of F whose magnitude is the change in F per unit distance.

- Hollywood