Formula $-$ valid or invalid?

In physics, during the derivation of a certain formula, it uses the definition that direction of area vector is perpendicular to the plane of the area. Now if I change the definition to "direction of area vector is not perpendicular to the plane of the area" and derive that formula, shall I reach the same formula? If not, the formula should be dependent on the definition that "direction of area vector is perpendicular to the plane of the area" which would make it an non-genuine formula. Am I right?

Edit @CR Drost

This is the derivation.

If we change the direction of area vector, will the final result change?

Edit@garyp

You said "Change to a symbol having a different but equally useful definition, and the formula will have to be modified".

In that case if we change the definition, then the final result i.e.potential energy should also change. But isn't this absurd? How can there be more than one potential energy for a system? There should indeed be one and only one potential energy for a system.

Edit @CR Drost

Do you mean while deriving a formula we define new symbols, compare them with each other and other formulas and get our required formula. Now if we change the definition of our symbols, we get another formula for the same quantity. Now how can a quantity have more than one formula? Which of the two formulas will be correct?

• The form of all formulas depends on the definitions of the symbols used. Change to a symbol having a different but equally useful definition, and the formula will have to be modified. What do you mean by "genuine"? – garyp Mar 27 '17 at 15:49

I don't know what you're referring to as a "non-genuine formula" or why you say something "is just a definition."

For example, we define that a "triangle" is a figure in flat 2D space made of 3 straight line segments, no two of which are parallel, that join end-to-end so that there are no "naked" endpoints of the segments. That statement "is just a definition."

We also define that if you choose one of its line segments as a "base" with length $b$, then draw successive lines parallel to this base going through the triangle, there is some line which is the most far away from the base while still intersecting the triangle (and it only intersects at one point). The distance between these two lines (base and the furthest away parallel that intersects the triangle) is the "height" $h$ of the triangle with respect to that base.

But now notice that based on these definitions the area of the triangle is always $A = \frac 12 bh.$ That seems to be a genuine formula, no? But it is based on these ideas which are very dependent on, say, which line segment you choose. But that's fine.

Similarly, the definition that we choose a normal vector $d\vec A$ to have magnitude equal to a differential area and direction perpendicular to the surface, is an important definition. For example it lets us define a current density field $\vec J$ such that the flow rate trickling through some little surface $d\vec A$ is given by the dot product $d\Phi = \vec J \cdot d\vec A.$ This definition then allows us to express the continuity equation that if the stuff which flows acccording to $\vec J$ has density $\rho$ then it is conserved if $$\frac{d\rho}{dt} = -\nabla \cdot J.$$ This is a genuine formula in physics, but of course it depends on the definition of $\vec J$ which depends on the definition that the direction of $d\vec A$ is perpendicular to the surface.

No, if the fact you use to derive a formula is true or if a formula is derived for a specific case then that formula is true for that case.

• can you elaborate? – N.G.Tyson Mar 27 '17 at 15:10
• The statement that "direction of area vector is perpendicular to the plane of the area" is just a definition. We can change the definition anytime we want. It cannot be a fact. – N.G.Tyson Mar 27 '17 at 15:13
• That's true , we derive the formula for this definition. A definition is a way to quantify something that has real applications. So the area vector you are talking about and the way it has been defined is because it is convenient and useful to do so. Now, the formula that you talk about needs to have a quantity that is perpendicular to the plane of area, why not the area vector? – Amit Hegde Mar 27 '17 at 15:20
• @faheemahmed400 It doesn't have to be a fact. It has to be consistent with how you derived the formula. In most formulas involving a dA vector, the vector is defined to be perpendicular to the plane of the area. It was likely chosen to make deviations work as easily as possible. Perpendicular relationships are also very important in physics, so it's reasonable that we end up defining the vector in that way. – JMac Mar 27 '17 at 16:41

While deriving a formula, definitions related to problem or that system are used as first step.Definition of area vector may seem not a general one ,but mainly it arises regarding concept of FLUX.here normal vector to a plane times area is pronounced as area vector that occurred in formula occasionally.So by changing the direction of this vector ,one may come up with something new formula,but that wouldn't be flux related concept for sure.In short,definition of area vector has arisen later after defining the concepts where u will see it.