I wish I had a more general answer, but I can give a specific example.
In order to work with circuits, differential equations (an area of calculus) must be used to understand and study the relationships between current, voltage, resistance, power, and work.
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in which field vector calculus is applied deeply
For Literally Everything.
Engineering careers, physicists, mathematicians, economists.
It is used to position an object in3D
The difference between Leibniz calculus to Newton calculus was that Leibniz developed Newton's calculus into the calculus we all know today. For instance, diffentiation and intergration, limits, continuity, etc. This type of calculus was the pure mathematics. On the otherhand, the calculus which Newton found was that used in physics, such as speed and velocity which helped with physics greatly. Today, calculus not only used in just mathematics or physics, but used in finance, as well as exploited in engineering.