Mathematics to me is a language we use to express observations. Just like any other language, depending on our vocabulary, we will be able to accurately represent our observations and depending on your fluency, be able to rephrase and understand how we may find a solution.
Let’s have a simple example.
For the capacitor, we need to know how to convert between differentials and integrals, to make v or i the subject on LHS. (Partly because I'm writing this on my phone and it’s easier to enter the differential format)
i(t)=c(dv/dt)
Without adequate knowledge of mathematics we would be unable to create those expressions for the components and therefore be unable to know what was going on in that circuit!
As far as the appearance of the mathematical terms is concerned, one could not distinguish between physical problems until the specification of certain parameters. In other words, the 'same' mathematical expression would be representative of various equivalent physical problems and the understanding of the derivation and the solution of one typical mathematical statement would allow for the subsequent understanding of the other equivalent problems. Such generality and interconnection between subjects which can only be made possible by the marriage between mathematics and engineering knowledge are 4 what is highly valued nowadays, especially with the advancement of computer technology. Quoting Trikha and Abang Abdullah (2004): "The IT revolution is in turn impacting on the construction industry in two major ways, the introduction of computer integrated construction and support for the construction of intelligent structures. The two developments making it imperative that countries, which have missed opportunities during the industrial revolution, take advantage of the IT revolution now. For the construction industry, computer integrated construction and intelligent buildings are the two current challenges which must be accepted with alacrity and enthusiasm’. Based on the above, it is enlightening to know that we, especially the engineers of developing countries, still have the chance to be at the frontier and becoming one of the world major players. This is because, whilst the last century was about fragmentation of scientists, engineers and sociologist etc., in their own specialized areas, IT revolution evoked the cross-bordering of specializations solely due to the availability of high-tech computers. For example, solving a set of non-linear of partial differential equations are no longer under the realms of the mathematicians; any engineering 'dons' could solve it with the help of commercial software such as Maple, MATLAB and MATHEMATICA. Such a privilege has caused the explosion of daring experimentations and unorthodox ideas all around the world which couldn’t even have been imagined fifty years ago. Such a privilege promotes the frequency and the efficiency of communications between various learned parties. Today's world headlines of bioengineering, for example, are simply possible because mathematicians, physicist, engineers and biologist talk in the same language, understanding each other. Therefore, any countries or societies failing to exploit such an opportunity would definitely fail in their endeavor to be the pioneer or frontier of anything.
Conclusions: On the question of how demanding is the 'engineering sense' hence the Engineering Mathematics, the answer is the single most important fact as it governs the whole discussion of to be or not to be. The answer is all spatial, time and personality dependent and initially, it would seem like a dilemma. Spatial wise, it depends on where the engineers working, i.e. company, country. If he or she works in accompany with less challenging projects (both in the practical and research aspects), we can suppose that this sense would be less demanding. The same argument applies to the country where the company is operating
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From @sazzad