What’s wrong with the mathematics of modern engineering?

In modern engineering, problems are generally solved with the cause and effect variables combined in parameters such as electrical resistance and heat transfer coefficient. It is mathematically unsound to solve problems with the cause and effect variables combined because it is generally easier to solve problems if the cause and effect variables are separated.

Solving problems with the variables combined vs separated

Equations (1) and (2) are identical.

(y/x) = 3 + (7/x) (1)

y = 3x + 7 (2)

The only difference between Eqs. (1) and (2) is that the variables x and y are combined in Eq (1), and separated in Eq (2).

Note that it is difficult to solve for x given y using Eq. (1), but quite easy using Eq. (2). For example, suppose that Eq. (1) must be used to determine the value of x at y = 12. Equation (1) cannot be solved directly because it contains the term (y/x). It must be solved iteratively, or by trial-and-error.

On the other hand, Eq. (2) is easy to solve for x given y (or y given x) because x and y are separated—ie they do not appear together in any term. If Eq. (2) is used to determine the value of x at y = 12, the problem is solved simply and directly, and the solution is obviously x = 5/3.

Similarly, if a chart is in the combined variable form y/x vs x, it must be read iteratively if y is given, and x is to be determined from the chart. But if a chart is in the separated variable form y vs x, it can be read directly if y is given and x is to be determined, and conversely.

“Heat transfer coefficient” combines cause and effect variables

“Heat transfer coefficient” is used in modern engineering. It is defined by Eq. (3):

q = h DT (3)

where q is heat flux, h is heat transfer coefficient, and DT is temperature difference.

Eq. (3) states that h is the ratio q/DT—ie h combines the cause and effect variables q and DT.

In modern engineering, Eq. (3) is “the defining equation for h”. The phrase “defining equation” is an oxymoron because equations describe behavior, and definitions do not. It was appropriate for Fourier to write Eq. (3) because he viewed it as a global description of behavior. In modern engineering, Eq. (3) is not a global description of behavior, and therefore it should not be written in the form of an equation. It should be written in the form of a definition—ie it should be written in the form “h ≡ q/ΔT”.

“Electrical resistance” combines cause and effect variables

“Electrical resistance” is used in modern engineering. It is defined by Eq. (4):

E = IR (4)

where E is electromotive force, I is electric current, and R is electrical resistance. Eq. (4) states that R is the ratio E/I—ie R combines the cause and effect variables E and I.

(It is interesting to note that Eq. (4) is generally referred to as “Ohm’s law”. However, the law expressed by Ohm in his 1827 treatise “The Galvanic Circuit Investigated Mathematically”, was not the homogeneous Eq. (4). It was the inhomogeneous equation

E = IL (5)

where L is the length of an equivalent copper wire of standard diameter.)

“Material modulus” combines cause and effect variables

“Material modulus” is used in modern engineering:

s = Ee (6)

where s is stress, e is strain, and E is material modulus. Equation (8) states that E is the ratio s/e—ie E combines the cause and effect variables stress and strain.

Why modern engineering is mathematically unsound

In modern engineering, problems are generally solved with the cause and effect variables combined in parameters such as electrical resistance and heat transfer coefficient. It is mathematically unsound to solve problems with the cause and effect variables combined because it is generally easier to solve problems if the cause and effect variables are separated.