in which field vector calculus is applied deeply

Hence the reason for why it is called Vector Calculus, Vector Calc. is simply an expansion in the calculus subject are in math. It deals with Taylor's Formula (in calc 2 you learn the taylor polynomial and the taylor series), theorems from Green, Gauss, and Stokes, and much more.

Richard H. Crowell has written:

'Calculus of vector functions' -- subject(s): Vector analysis

'Calculus with analytic geometry' -- subject(s): Analytic Geometry, Calculus

H. K. Nickerson has written:

'Advanced calculus, by H.K. Nickerson, D.C. Spencer and N.E. Steenrod' -- subject(s): Calculus, Vector analysis

'Advanced calculus' -- subject(s): Calculus, Vector analysis

Electromagnetic fields, gravitational fields and fluid flow. If you are an engineer you will come across vector calculus to handle three dimensional space.