Vector Calculus Concepts and Applications Explained

In Mathematics, calculus refers to the branch which deals with the study of the rate of change of a given function. Calculus plays an important role in several fields like engineering, science, and navigation. Usually, calculus is used in the development of a mathematical model for getting an optimal solution. You know that calculus is classified into two different types which are known as differential calculus and integral calculus. However, you might not be aware of vector calculus. In these vector calculus pdf notes, we will discuss the vector calculus formulas, vector calculus identities, and application of vector calculus. Let us first take a look at what is vector differential calculus in these vector calculus notes.

Vector Calculus Definition

Vector calculus 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 is 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.

1. 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 is 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.

2. 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

1. Fundamental Theorem of the Line Integral

Consider F=▽f and a curve C that has the endpoints A and B.

Then you would get

\[\int cF .dr = f(B) -f(A)\]

2. Circulation Curl Form

According to the Green’s theorem,

\[\iint_{D}\left ( \frac{\partial Q}{\partial x} \right )- \left ( \frac{\partial P}{\partial y} \right )dA = \oint CF. dr\]

According to the Stoke’s theorem,

\[\iint_{D}\bigtriangledown \times F.nd\sigma = \oint CF. dr\]

Here, C refers to the edge curve of S.

3. Flux Divergence Form

According to the Green’s theorem,

\[\iint_{D}\bigtriangledown .F dA = \oint CF. nds\]

According to the Divergence theorem,

\[\int \int \int_{D}\triangledown .FdV\] = ∯  SF. ndσ

Vector Calculus Identities

Let us learn about the different vector calculus identities. The list of the vector differential calculus identities is given below.

1. Gradient Function

1. \[\vec{\bigtriangledown}(f+g) =  \vec{\bigtriangledown}f + \vec{\bigtriangledown}g\]

2. \[\vec{\bigtriangledown}(cf) =  c\vec{\bigtriangledown}f\], for a constent c

3. \[\vec{\bigtriangledown}(fg) =  f\vec{\bigtriangledown}g + g\vec{\bigtriangledown}f\] 

4. \[\vec{\bigtriangledown}(\frac{f}{g}) =  \frac{g\vec{\bigtriangledown}f-f\vec{\bigtriangledown}g}{g^{2}}\] at the point \[\vec{x}\] where g \[(\vec{x}) \neq 0\]

5. \[\vec{\bigtriangledown} (\vec{F}.\vec{G}) = \vec{F}\times (\vec\bigtriangledown\times\vec G  )- (\vec\bigtriangledown\times\vec F  )\times \vec G + (\vec G .\vec{\bigtriangledown})\vec F + (\vec{F}.\vec{\bigtriangledown})\]

2. Divergence Function

1. \[\vec{\bigtriangledown} (\vec{F}+\vec{G}) = \vec{\bigtriangledown}.\vec{F} + \vec{\bigtriangledown}.\vec{G}\]

2. \[\vec \bigtriangledown.(c\vec{F)} = c \vec{\bigtriangledown .\vec F }\]

3. \[\vec \bigtriangledown.(f\vec{F)} = f \vec{\bigtriangledown .\vec F }+ \vec F .\vec \bigtriangledown\]

4. \[\vec \bigtriangledown.(\vec{F}\times \vec{G}) = \vec{G}. (\vec{\bigtriangledown \times \vec{F}})-\vec{F}.(\vec{\bigtriangledown \times \vec{G}})\]

3. Curl Function

1. \[\vec \bigtriangledown\times (\vec{F}+\vec{G}) = \vec{\bigtriangledown \times \vec{F}}+ \vec{\bigtriangledown }\times \vec{G}\]

2. \[\vec \bigtriangledown\times (c\vec{F)} = c\vec{\bigtriangledown } \times \vec{F}\], for a constant c

3. \[\vec \bigtriangledown\times (f\vec{F)} = f\vec{\bigtriangledown } \times \vec{F} + \vec{\bigtriangledown }f\times \vec{F}\]

4. \[\vec \bigtriangledown\times (\vec{F}\times \vec{G}) = \vec{F}.(\vec{\bigtriangledown . \vec{G}})-(\vec{\bigtriangledown }\vec{F})\vec{G} + (\vec{G}. \vec{\bigtriangledown })\vec{F} -(\vec{F}.\vec{\bigtriangledown }  )\]

4. Laplacian Function

1. \[\vec{\bigtriangledown ^{2}}(f+g) = \vec{\bigtriangledown ^{2}}f + \vec{\bigtriangledown ^{2}}g\]

2. \[\vec{\bigtriangledown ^{2}}(cf) = c\vec{\bigtriangledown ^{2}}f\], for a constant c

3. \[\vec{\bigtriangledown ^{2}}(fg) = f\vec{\bigtriangledown ^{2}}g + 2\vec{\bigtriangledown f}.g + g \vec{\bigtriangledown ^{2}}\]

5. Degree Two Function

1. \[\vec{\bigtriangledown }.(\vec{\bigtriangledown \times \vec{F}})\] = 0

2. \[\vec{\bigtriangledown }\times (\vec{\bigtriangledown f})\] = 0

3. \[\vec{\bigtriangledown }.(\vec{\bigtriangledown  f}\times\vec{\bigtriangledown  g} ) = 0\]

4. \[\vec{\bigtriangledown }.(f\vec{\bigtriangledown  g}- g\vec{\bigtriangledown f}) = f \vec{\bigtriangledown ^{2}}g - g \vec{\bigtriangledown ^{2}}f\]

5. \[\vec{\bigtriangledown }\times (\vec{\bigtriangledown \times \vec{F}}) = \vec{\bigtriangledown } (\vec{\bigtriangledown . \vec{F}}) - \vec{\bigtriangledown ^{2}}\]