An engineering career requires you to understand how motion, even minute, affects matter. This would require a good knowledge of higher calculus. Take as many courses in mathematics that you can and make sure you have an excellent grasp of differential calculus. Eventually understand Newton and Liebnitz and their theories.
in which field vector calculus is applied deeply
CALCULUS
For Literally Everything.
These are the general math courses in an undergraduate program of Mechanical Engineering. Actually, these are also the math courses required in ANY undergraduate Engineering curriculum: Algebra Trigonometry Analytic Geometry Differential Calculus Integral Calculus Mutivariable Calculus Differential Equations
Because calculus is lots of fun! Also because it is useful in science and engineering.
Vector calculus is applied in electrical engineering especially with the use of electromagnetics. It is also applied in fluid dynamics, as well as statics.
Not much. In any case, you won't need advanced math (like trigonometry, algebra, calculus, logarithms...) which you only use in science and engineering careers.
Calculus was created to prove physics which defines the laws of nature.
Euler didn't discover calculus. He made major contributions to calculus throughout his career, but the foundations of calculus were put forth by Newton and Leibniz.
Mechanical engineering usually deals with forces and their effects on materials. Forces are vectors and so, to study their effects you need to use vector calculus.
Quality control is an engineering career that starts with a q.
Engineering careers, physicists, mathematicians, economists.