How to visualize the maths behind a formula?

[ward0]

TL;DR Summary

Most of physics is expressed through mathematical notation. When there's lots of division, multiplication, exponentiation of various terms, I struggle to find meaning in equations

I just finished my freshman year in computer engineering at my college, and after some physics and materials science classes I noticed a pattern. Everybody—at least in my class, I don't about the rest of the world—is just memorizing formulas, myself included. I'm really trying to not just memorize then and instead understand them, and I'm struggling a lot.

Formulas are just a way to express ideas through mathematical notation, I (think) know this much. When they're simple (e.g., the average velocity formula), I can grasp the meaning of an addition, a subtraction, but I even struggle finding meaning when there's division and multiplication sometimes, let alone exponentiation.

For example sake, here's the diffusion coefficient formula for solids (Callister, 2010):

I don't a have a clue on what it means. I can explain each term independently, I know its application, but I don't know their relation, and this happens in almost every formula presented to me. No one actually bothered explaining this to me, and I just realized now that I pass through formulas and never actually understand them.
Btw I (think, at least) am ok on pure maths, I understood integrals and differentiation without problem, I also know what they mean graphically.

I came here for help, suggestions on reading, articles, even videos, anything. I really want to understand how to visualize these equations and know why it has the terms it has and what they're doing there. If anyone could help me I'd be very grateful.

 

[fresh_42]

Not sure whether this is actually a mathematical rather than a physics question.

The first thing you do is grasp the main information: In your example you have $D=D_0e^\alpha$, which means that $D$ depends exponentially on $\alpha$. Assuming $D_0> 0$ this automatically means a graph like this:

One way to understand is to use a graphic program and play with the constants $D_0$ and $\alpha$ to see how the curve changes. Now $\alpha$ is the next part of the equation. What is here the main information? $Q_d$ and $R$ are constants, so we have basically $\alpha \sim \dfrac{1}{T}$.

This changes our initial graph, since we now have $D \sim e^{1/T}$, a quantity which quickly (exponentially) decreases with (linearly) increasing temperature:

This reminds of a radioactive decay over time.

Conclusion: The diffusion coefficient decreases exponentially with increasing temperature. The specific curves depend on the gas involved.

I only used Wikipedia to look up the meaning of your variables. The rest was just what the formula told me. Hence I learned something about the diffusion coefficient without even knowing what is measured to determine it.

Edit: I've missed the minus in the exponent, which makes the function look like the one in the post below!

 

[PeroK]

Summary:: Most of physics is expressed through mathematical notation. When there's lots of division, multiplication, exponentiation of various terms, I struggle to find meaning in equations

For example sake, here's the diffusion coefficient formula for solids (Callister, 2010):
View attachment 254296
I don't a have a clue on what it means. I

That's not a particularly easy equation to grasp. First, however, you have to look at the negative exponential, which comes up a lot in physics. Often it involves the time parameter. For example:

$X = X_0 \exp(-\alpha t)$

That says that a quantity $X$ starts at $X_0$ at $t = 0$ and decays to zero over time. How quicky it decays depends on $\alpha$. A larger $\alpha$ gives a faster decay. I'm assuming here that $\alpha$ is a positive quantity.

Note also that in a formula like this the quantity $\alpha t$ must be dimensionless. I.e. alpha must have the units of inverse time, i.e. frequency or angular frequency.

For example, you see this also in simple harmonic motion: $x = A\cos(\omega t)$, where $\omega$ is the angular frequency with units of inverse time and $\omega t$ is dimensionless.

Your equation is more complicated because of the $1/T$. Assuming $T$ is the important variable, and $Q_d/R$ is some constant of the system in question, $D$ has starts at a zero value at absolute zero ($T=0 \ K$) and increases towards a hypothetical maximum of $D_0$ as $T \rightarrow \infty$. I.e. you have an equation of the form:

$D = D_0 \exp(-\frac{\alpha}{T})$

Note that I've assumed that $\alpha$ is positive here.

This looks like the right-hand side here, with $y = D$ and $x = T$:

Note that $D$ will increase fairly rapidly to something close to $D_0$ and then flatten out.

 

[PeroK]

Not sure whether this is actually a mathematical rather than a physics question.

The first thing you do is grasp the main information: In your example you have $D=D_0e^\alpha$, which means that $D$ depends exponentially on $\alpha$. Assuming $D_0> 0$ this automatically means a graph like this:

View attachment 254297

One way to understand is to use a graphic program and play with the constants $D_0$ and $\alpha$ to see how the curve changes. Now $\alpha$ is the next part of the equation. What is here the main information? $Q_d$ and $R$ are constants, so we have basically $\alpha \sim \dfrac{1}{T}$.

This changes our initial graph, since we now have $D \sim e^{1/T}$, a quantity which quickly (exponentially) decreases with (linearly) increasing temperature:

View attachment 254298

This reminds of a radioactive decay over time.

Conclusion: The diffusion coefficient decreases exponentially with increasing temperature. The specific curves depend on the gas involved.

I only used Wikipedia to look up the meaning of your variables. The rest was just what the formula told me. Hence I learned something about the diffusion coefficient without even knowing what is measured to determine it.

I would have assumed that the quantity $Q_d$ is positive, in which case we have a different graph. See above.

 

[fresh_42]

I would have assumed that the quantity $Q_d$ is positive, in which case we have a different graph. See above.

Damn, I've missed the minus sign ...

 

[Stephen Tashi]

Formulas are just a way to express ideas through mathematical notation, I (think) know this much. When they're simple (e.g., the average velocity formula), I can grasp the meaning of an addition, a subtraction, but I even struggle finding meaning when there's division and multiplication sometimes, let alone exponentiation.

I think you are commenting about understanding the significance or reason behind a formula rather than understanding how to execute the formula to produce a numerical result.. You are correct that formulas state mathematical ideas. However, the significance of a formula cannot be appreciated by seeking a verbal explanation for why each arithmetical operation in the formula occurs. Taking that approach is analogous to the approach of looking at each individual word in a sentence and combining their definitions to produce the meaning of the sentence as a whole. This works well in many disciplines, but it does not work in mathematics.

Btw I (think, at least) am ok on pure maths, I understood integrals and differentiation without problem, I also know what they mean graphically.

To understand the frequent occurence of exponentials in formulas, you need to understand differential equations. Have you studied them? Very often, understanding a formula in physics involves understanding why a certain differential equation is a model for a physical phenomena and then understanding why certain mathematical functions are solutions to that differential equation.

 

[symbolipoint]

ward0,
Maybe your complaint is because the instruction is on engineering and applications, and not on understanding? I could not tell you for certain. Answering your question is not easy. Stephen Tashi gave a good response.

 

[scottdave]

One thing I like to do: think about how a particular variable, such as T affects the value of the left-hand side, when it approaches extremes.

• What happens when T gets very small (approaches zero)?
• Are there limits on what values that T can take?
• What happens as T gets very large?

Answering these can help you to somewhat understand what is going on between T and D. No graphing software required.
What about the other quantities?

Another thing to think about, the argument of an exponential or logarithm is nearly always dimensionless. For example if you have a variable inside a log that you don't know what it means, you should be able to figure out what dimensions it needs to be to cancel out the others.