Vector Calculus: Understanding Flux

Once you understand flux intuitively, you don’t need to memorize equations. The formulas become “obvious” dare I say. However, it took a lot of effort to truly understand that:

This is a companion discussion topic for the original entry at

how to measure the leakage flux…ie the flux that is left unused as in case of a electrical machine…?

Hi Singaram, unfortunately I don’t know much about leakage flux for electrical devices – it appears to be more of an engineering problem.

From a theoretical standpoint, you could surround your transformer with a sphere or cube, and measure the flux passing through each circuit (assuming you knew the magnetic field vector).

nice but like to see more math

whats the relation between dot product (.) and cross (x) products? how are they related to vectors and scalars?

Hi Gerard, the dot product is the amount one vector “pushes” in the direction of the other. If they are perpendicular, there is zero push. If they are parallel, there is maximum push. The dot product gives a single number, a scalar.

The cross product is a way to find the “area” spanned by two vectors. In this case, you want them to be perpendicular (to make the largest polygon). If they are parallel, there is no “area” between them. The cross product gives another vector, which is perpendicular to the input vectors.

Great stuff. I am one step closer to understanding flux as it relate to Quantum Mechanics.

Thanks Tasha, glad you enjoyed it.

thank you for your scalar/vector explanation. another puzzle for me has been the bernoulli theorem if it’s valid how does a plane fly when itis inverted? wouldn’t it tend to be forced downward and crash if the higher pressure on the wing is on top?

@Gerard: Thanks for the comment. Unfortunately, I don’t know much about Bernoulli’s theorem, but I’ve read that the angle of attack is as or more important than the exact wing shape (thus flying upside down works with the correct angle of attack).

thanks kalid for the bernoulli answer. how about another mystery – if the purpose of finding solutions to ordinary differential equations is to find a function answer rather than a numerical one, do partial diff eqs find partial function answers?

[…] Div, Grad, Flux and Curl (if you already know vector calculus) […]

A minor point: In the Flux Factors section the analogy with a sail, “like a sail facing directly into the wind”, is awkward.
My first reaction was “Huh? That would be zero flux”. And here’s why…

If you’re picturing a bermuda-rigged ship, to say a sail is “directly into the wind”, doesn’t really make sense. If the ship were directly into the wind the sail is catching no wind (and that’s what I first pictured).

(If you were picturing a square-rigger instead, the sails would be backed. That’s just weird.)

So for clarity, drop the analogy, or talk about a ship running before the wind - maximum flux in both Bermudas and Square-riggers.

Great explanation, thanks a lot!

[…] Vector Calculus: Gradient, Flux, Divergence, Curl & Circulation […]

Thanks that really helped.
But I have 2 problems, can you please help me out?

  1. You said "would you rather have a handful of $5 or $20 bills “flux” into your bank account? Would you rather have a big or little banana come your way? No need to answer that one."
    we’ll wish to have $5 “flux” into our account as flux means passing thru, so if we had $20 fluxed into our account, we’ll be paying more, right?
    So, is it what you meant?
    But in case of bananas, you said “come your way” but are you considering banana(s) as profit, if that’s so we’ll want the big one other wise the smaller one.
    So what do you mean by this?

Okay, sorry I pressed enter by mistake.
Here’s the second thing I was to say.

  1. You said "Flux is a total, and is not “per unit area” or “per unit volume”. Flux is the total force you feel, the total number of bananas you see flying by your surface"
    But later you said “your field could represent bananas-per-second, in which case you’d get the bananas-per-second crossing your surface. The units of flux depend on the units of your vector field.”

Now what’s that? You are contradicting yourself and what’s more you are confusing us ( or atleast me ).
Please make it clear. Is there a unit or not?

My book says :
The surface integral (integral sign and then A.dS with bars over them to show that they are vectors)represent ‘flow of flux’ of vector field A over surface S. Say, if A = pV (A and V vectors) where p is density and V is velocity of fluid then surface integral (the integral) represent amount of fluid flowing through given surface in unit time.

You see, it says “flow of flux” now what does that means?
Also it says “in unit time” so does flux indeed have units, or not as you first said (then you contradicted yourself).
Please explain, I m very confused.

@aaryan: Thanks for the comments –

  1. Yes, in both cases (bananas and dollars) the intent for the analogy is for both types of flux to be “good”. So I mean to say that bigger is better in both cases.

  2. Good question – I might need to go back and clarify. Flux is defined as the total impact of the field over the entire surface, and not a “piece by piece” impact.

The unit is Force * Area, so you’d have to multiply out the units for whatever force and area represent in your particular situation. In some cases, the “Force” will represent a velocity (m/s), work done, or an amount of something passing through a single point (bananas per second, through this exact point). I’ll need to clarify but flux represents a “multiplication” of force across a certain area. The units of that multiplication will depend on the problem, but in general flux is the total impact of the force on the entire area (not the impact of the force on one particular point).

Thank you for this explanation. I am taking bioelectrics several years after calc 3 and was looking for a refresher. This is better than my calc 3 book.

Feynman once said about some classmates pondering the mysteries of a French curve, “they didn’t even know what they knew.” Thanks for helping people know what they know.

@Wendy: You’re more than welcome, really glad it helped! I like Feynman, I hadn’t heard that quote – there are so many things that we know on a surface level but don’t understand. Every day I’m seeing more things which I thought I “knew” :). Thanks for writing.