Vector Integration
Vector Calculus Formulas
Some of the important vector calculus formulas are given below:
From fundamental theorems, we take,
F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k
- Fundamental Theorem of Line Integral
If F = ∇f and curve C has endpoints A and B, then
∫c F. dr= f(B) − f(A).
- Circulation-Curl Form
Green’s Theorem
∫∫D (∂Q /∂x) – (∂P/ ∂y)dA = ∮C F· dr
Stokes’ Theorem
∫∫D ∇ × F · n dσ = ∮C F· dr, where C is the edge curve of S
- Flux – Divergence Form
Green’s Theorem
∫∫D ∇· F dA = ∮C F · n ds
Divergence Theorem
∫∫∫D ∇· F dV = ∯S F · n dσ
Vector Calculus Definition
Vector Calculus, also known as vector analysis, deals with the differentiation and integration of vector field, especially in the three-dimensional Euclidean space. Vector fields represent the distribution of a vector to each point in the subset of space. In Euclidean Space, the vector field on a domain is represented as a vector-valued function that compares the n-tuple of real numbers to each point on the domain. Vector analysis is an analysis which deals with the quantities that have both magnitude and direction. Vector calculus deals with two integrals such as line integrals and surface integrals.
Vector Calculus Applications
Vector Calculus plays an important role in the different fields such as;
- Used in heat transfer
- Navigation
- Sports
- Three-dimensional geometry
- Partial differential equation