# What does a gradient mean in physics?

I'm a physics high school student and have learned about the term 'gradient' regarding a few situations, such as pressure gradients and temperature gradients.

But what does this really mean? What is the physics meaning of gradient? I know that the pressure gradient is $\frac{dP}{dx}$ and the temperature gradient is $\frac{d\theta}{dx}$. If we take the $dx$, for instance, as an extremely small number, then the gradient approaches towards a very large value. What does this imply? Please explain in simple language!

• Do not forget that P is a function of x, so if P(x) is continuous, then if dx is small then dP is small too, the ratio is independent of the value of dx if dx is small enough.en.wikipedia.org/wiki/Derivative
– user126422
Commented Feb 24, 2017 at 3:46
• The best explanation of gradient on the internet can be found here betterexplained.com/articles/…
– Paul
Commented Feb 24, 2017 at 6:07
• Appreciate the edits Yashas :-) I made a lazy choice not to, and that is on me. Commented Feb 24, 2017 at 7:39

I struggled with the concept myself even in later calculus (where 2 and 3-dimensional gradient operators are developed)... which is a real problem when a meteorology major!

But one day it just dawned on me that it's as simple as it sounds. It's the rate of difference.

As Gary mentioned, in one dimension, a gradient is the same as a slope.
As you indicated, in $\frac{dP}{dx}$, if you decrease $dx$, it would seem mathematically to be pushing the result to larger values. But in actuality, when you consider a smaller $dx$ (distance), you also will consequently see a smaller change in the property of interest (pressure in this case).
It's exactly like working with a line... if you have a slope of $2$, you have a slope of $2$ regardless of the scale you look at it on. If you look at a smaller $x$ change in the line, say $dx = 0.01 \ldots$ then the $y$ changes follow suit, and $dy$ is just $0.02$. They vary together. $\frac{dy}{dx}$ is a ratio.

It also helped me to step back and reconsider the concept/meaning/definition of derivatives again. Remember, $dP/dx$ is just $\frac{\Delta P}{\Delta x}$. Except applied at an "instantaneous" spot. Typically we still calculate $\frac{dP}{dx}$ observationally using $\frac{\Delta P}{\Delta x}$. It's just that in the real world, things usually don't vary consistently (linearly); some spots have a "quicker" change than others, such the temperature gradient across a cold front. So $\frac{\Delta P}{\Delta x}$ doesn't take into account variations in the slope itself, so we transition to $\frac{dP}{dx}$ ideally for perfection. Even when we're incapable of obtaining such a perfect value outside of theoretical/derivation circumstances. But it's still the same basic idea either way: how "quickly" the value ($P$) changes over a given direction ($x$).

It was frustrating to wrap my head around. But spend enough time staring at a weather map or thinking about it as you climb complex terrain, and I think you'll really understand that we don't pick $dx$, but that $\frac{dy}{dx}$ is truly just one quantity just like a slope, linking the two variables together. And I'm confident it will suddenly click innately what a gradient means! (even in 2-D or 3-D... that just adds a direction in which the gradient changes)

Gradient refers to how steep a line is, which is basically the slope. $\frac{dP}{dx}$ and $\frac{d\theta}{dx}$ are basically the derivative of a function, i.e its slope.

The easiest way to determine slope is to graph the function, then observe the x coordinate of a point on the graph and its respective y coordinate. If the y coordinate is increasing as the x coordinate is increasing, the slope /gradient is said to be positive at that point and if the y coordinate is decreasing as the x coordinate is increasing, the slope is said to be negative.

• In the pressure gradient thing,I am still confused about it's physical meaning.It says "The change in pressure per unit change in length is called pressure gradient".Does this hold any physical significance or is this just about maths(which I am pretty bad in).Why is pressure gradient used in deriving Newton's law of viscosity?Please help Commented Feb 24, 2017 at 5:15
• First of all gradient is not only restricted to pressure or any particular topic in physics. It's a general concept. "Why is pressure gradient used in deriving Newton's law of viscosity? " is a completely different answer! If you have a random line graphed in an x, y plane how will you determine the slope between 2 points? Commented Feb 24, 2017 at 5:17

What I know from the basic is that..

Gradient is a slope of a curve in a graph at specific points on the curve. If a curve is linear then gradient remain constant. If it is a curve then gradient vary.

A gradient of something is simply a quantity that changes over distance.

Does the pressure change over distance? Pressure gradient

Does the temperature change over distance? Temperature gradient.

Does the height of the road change over distance? Gradient (this is so common that we leave out the height).

The derivative $$\frac{dy}{dx}$$ in its general sense is used to describe how a quantity $$y$$ changes as we vary $$x$$. When we 'vary time' we usually call the derivative a rate. When we 'vary space' we usually call it a gradient. In other words, the gradient is just a specific name for the derivative.

If we take the dx , for instance, as an extremely small number, then the gradient approaches towards a very large value. What does this imply?

This is not entirely correct. Let's look for example at a pressure gradient $$\frac{dP}{dx}$$. When we move by a small amount $$\Delta x$$, the pressure will change by an amount $$\Delta P$$. When $$\Delta x$$ gets very small, $$\Delta P$$ will also get very small. How fast? Well, $$\Delta P$$ will usually get smaller in such a way that the fraction $$\frac{\Delta P}{\Delta x}$$ approaches a constant value. If it does, we call this value the derivative $$\frac{dP}{dx}$$. Here is a hypothetical example where $$\frac{dP}{dx}=2$$.

$$\Delta x$$ $$\Delta P$$ $$\frac{\Delta P}{\Delta x}$$
0.1 0.231 2.31
0.01 0.02051 2.051
0.001 0.0020109 2.0109
0.0001 0.0002000763 2.000763

When does a function have a derivative? We call a function that has a derivative everywhere smooth. A function is smooth when it doesn't have large jumps or jagged edges. A smooth function will always look like a straight line if you zoom in enough. Things like temperature and pressure are almost always smooth. A counterexample would be the value of a stock in the stock market. The value of stocks looks jagged and keeps looking jagged even if you zoom in on a particular point.