What is Vector Calculus?

Vector Calculus is a branch of mathematics that deals with the operations of calculus i.e. differentiation and integration of vector field usually in a 3 Dimensional physical space also called Euclidean Space. The applicability of Vector calculus is extended to partial differentiation and multiple integration. Vector Field refers to a point in space that has magnitude and direction. These Vector Fields are nothing but Vector Functions. Vector calculus is also known as vector analysis.

The vector fields are the vector functions whose domain and range are not dimensionally related to each other. The branch of Vector Calculus corresponds to the multivariable calculus which deals with partial differentiation and multiple integration. This differentiation and integration of vector is done for a quantity in 3D physical space represented as R³. For n-dimensional space, it is represented as Rⁿ.

Read in Detail: Calculus in Maths

Vector Calculus Definition

Vector calculus, also known as vector analysis or vector differential calculus, is a branch of mathematics that deals with vector fields and the differentiation and integration of vector functions

Vector Calculus often called Vector Analysis deals with vector quantities i.e. the quantities that have both magnitude as well as direction. Since we know that Vector Calculus deals with differentiation and integration of functions, there are three types of integrals dealt with in Vector Calculus that are 

• Line Integral
• Surface Integral
• Volume Integral

Let’s learn about these integrals in detail.

Line Integral

Line Integral in mathematics is the integration of a function along the line of the curve. The function can be a scalar or vector whose line integral is given by summing up the values of the field at all points on a curve weighted by some scalar function on the curve. Line Integral is also called Path Integral and is represented by Φ = ∫_Lf. Line Integral has got its application in physics. For Example, Work Done by Force is along a path given as W = ∫_LF(s).ds because we know that work done is given as the product of force and distance covered.

Surface Integral

Surface Integral in mathematics is the integration of a function along the whole region or space that is not flat. In Surface integral, the surfaces are assumed of small points hence, the integration is given by summing up all the small points present on the surface. The surface integral is equivalent to the double integration of a line integral. Surface Integral has got its application in Electromagnetism and many more branches of physics where the vector function is spread over the surface. Surface Integral is represented as ∬_sf(x,y)dA.

Learn more about Double Integral.

Volume Integral

A volume integral, also known as a triple integral, is a mathematical concept used in calculus and vector calculus to calculate the volume of a three-dimensional region within a space. It is an extension of the concept of a definite integral in one dimension to three dimensions.

Mathematically, the volume integral of a scalar function f(x, y, z) over a region R in three-dimensional space is denoted as:

[Tex]\bold{\iiint_R f(x, y, z) \ dV} [/Tex]

Where 

• dV represents an infinitesimal volume element, and 
• Integral is taken over region R.

Vector Calculus in maths is a sub-division of Calculus that deals with the differentiation and integration of Vector Functions. We already know that Calculus is a branch of mathematics that deals with the rate of change of a function concerning another function. There are two major divisions of Calculus namely, Differential Calculus and Integral Calculus.

The branch of Differential Calculus deals with the process of finding derivatives or differentiation of functions while Integral Calculus deals with finding the antiderivative of a function whose derivative is given. In this article, we will learn in detail about Vector Calculus which is a lesser-known branch of calculus, and the basic formulas of Vector Calculus.

In this article, you are going to read everything about what is vector calculus in engineering mathematics, vector calculus formulas, vector analysis, etc.