Table of Contents
Vector Calculus Formulas
Vector Calculus – Definition: Vector calculus is the study of the calculus of vectors and vector fields. It is a powerful tool for solving problems in physics and engineering.
The basic vector calculus formulas are:
• Derivative of a vector:
dv = lim dx→0 (v + dx) − v
dv = (v · dx) + (dx · dv)
• Gradient of a vector:
∇v = (v · ∇)v
• Divergence of a vector:
div v = (v · div)v
• Curl of a vector:
curl v = (v · curl)v
Vector Calculus Definition
Vector calculus is a branch of mathematics that deals with the properties and behavior of vectors, vector fields, and tensors in three-dimensional space. It incorporates the principles of calculus and linear algebra to investigate the properties of vectors and vector fields, and to find solutions to problems involving them.
Vector calculus is used in a wide range of applications, including physics, engineering, and mathematics. It is particularly important in the study of fluids and electromagnetism.
Vector Calculus Definition
It is also known as vector analysis which deals with the differentiation and the integration of the vector field in the three-dimensional Euclidean space. Vector fields represent the distribution of a given vector to each point in the subset of the space. In the Euclidean space, the vector field on a domain represented in the form of a vector-valued function which compares the n-tuple of the real numbers to each point on the domain.
Vector analysis is a type of analysis that deals with the quantities which have both the magnitude and the direction. Vector calculus also deals with two integrals known as the line integrals and the surface integrals.
Line Integral
According to vector calculus, the line integral of a vector field is known as the integral of some particular function along a curve. In simple words, the line integral said to be integral in which the function that is to be integrated is calculated along with the curve. You can integrate some particular type of the vector-valued functions along with the curve. For example, you can also integrate the scalar-valued function along the curve. Sometimes, the line integral is also called the path integral, or the curve integral or the curvilinear integrals.
Surface Integral
In calculus, the surface integral is known as the generalization of different integrals to the integrations over the surfaces. It means that you can think about the double integral being related to the line integral. For a specific given surface, you can integrate the scalar field over the surface, or the vector field over the surface.
Vector Calculus Formulas
Let us now learn about the different vector calculus formulas in this vector calculus pdf. The important vector calculus formulas are as follows:
From the fundamental theorems, you can take,
F(x,y,z)=P(x,y,z)i+Q(x,y,z)j+R(x,y,z)k
Fundamental Theorem of the Line Integral
Consider F=▽f and a curve C that has the endpoints A and B.
Then you would get
∫cF.dr=f(B)−f(A)∫cF.dr=f(B)−f(A)
Circulation Curl Form
According to the Green’s theorem,
∬D(∂Q∂x)−(∂P∂y)dA=∮CF.dr∬D(∂Q∂x)−(∂P∂y)dA=∮CF.dr
According to the Stoke’s theorem,
∬D▽×F.ndσ=∮CF.dr∬D▽×F.ndσ=∮CF.dr
Here, C refers to the edge curve of S.
Flux Divergence Form
According to the Green’s theorem,
∬D▽.FdA=∮CF.nds∬D▽.FdA=∮CF.nds
According to the Divergence theorem,
∫∫∫D▽.FdV∫∫∫D▽.FdV
