Vector Calculus

Vector Calculus Formulas

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 the vector calculus. In these vector calculus pdf notes, we will discuss about 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 the 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.

  1. Surface Integral

In calculus, the surface integral is known as the generalisation 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 \[F = \triangledown f\] and a curve C that has the endpoints A and B.

Then you would get

\[\int _{c} F. dr = f(B) - f(A)\]

  1. Circulation Curl Form

According to the Green’s theorem,

 \[\iint _{D} (\frac{\partial Q}{\partial x}) - (\frac{\partial P}{\partial y}) dA = \oint C F. dr\]

According to the Stoke’s theorem,

\[\iint _{D} \triangledown\times F . n d\sigma = \oint C F \cdot dr\]

Here, C refers to the edge curve of S.

  1. Flux Divergence Form

According to the Green’s theorem,

\[\iint _{D} \triangledown . F dA = \oint C F \cdot n ds\]

According to the Divergence theorem,

\[\iiint _{D} \triangledown \cdot F dV\] = \[SF \cdot n d\sigma\]

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. \[\overrightarrow{\triangledown} (f + g) = \overrightarrow{\triangledown} f + \overrightarrow{\triangledown} g\].

  2. \[\overrightarrow{\triangledown} (cf) = c \overrightarrow{\triangledown} f\], for a constant c.

  3. \[\overrightarrow{\triangledown} (fg) = f\overrightarrow{\triangledown} g + g\overrightarrow{\triangledown} f\].

  4. \[\overrightarrow{\triangledown} (\frac{f}{g}) = \frac{(g \overrightarrow{\triangledown} f - f \overrightarrow{\triangledown} g)}{g^{2}}\] at the points \[\overrightarrow{x}\] where \[g(\overrightarrow{x}) \neq 0\].

  5. \[\overrightarrow{\triangledown} (\overrightarrow{F} \cdot \overrightarrow G) = \overrightarrow{F}\times (\overrightarrow{\triangledown}\times \overrightarrow{G}) - (\overrightarrow{\triangledown}\times \overrightarrow{F})\times \overrightarrow{G} + (\overrightarrow{G} \cdot \overrightarrow{\triangledown}) \overrightarrow{F} + (\overrightarrow{F} \cdot \overrightarrow{\triangledown})\]

2. Divergence Function

  1. \[\overrightarrow{\triangledown} \cdot (\overrightarrow{F} + \overrightarrow{G}) = \overrightarrow{\triangledown} \cdot \overrightarrow{F} + \overrightarrow{\triangledown} \cdot \overrightarrow{G}\]

  2. \[\overrightarrow{\triangledown} \cdot (c\overrightarrow{F}) = c \overrightarrow{\triangledown} \cdot \overrightarrow{F}\], for a constant c.

  3. \[\overrightarrow{\triangledown} \cdot (f\overrightarrow{F}) = f\overrightarrow{\triangledown} \cdot \overrightarrow{F} + \overrightarrow{F} \cdot \overrightarrow{\triangledown}\]

  4. \[\overrightarrow{\triangledown} \cdot (\overrightarrow{F}\times \overrightarrow{G}) = \overrightarrow{G} \cdot (\overrightarrow{\triangledown}\times \overrightarrow{F}) - \overrightarrow{F} \cdot (\overrightarrow{\triangledown}\times \overrightarrow{G})\]

3. Curl Function

  1. \[\overrightarrow{\triangledown}\times (\overrightarrow{F} + \overrightarrow{G}) = \overrightarrow{\triangledown}\times \overrightarrow{F} + \overrightarrow{\triangledown}\times \overrightarrow{G}\]

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

  3. \[\overrightarrow{\triangledown} \times (f\overrightarrow{F}) = f\overrightarrow{\triangledown} \times \overrightarrow{F} + \overrightarrow{\triangledown}f \times \overrightarrow{F}\]

  4. \[\overrightarrow{\triangledown}\times (\overrightarrow{F} \times \overrightarrow G) = \overrightarrow{F}\cdot (\overrightarrow{\triangledown}\cdot \overrightarrow{G}) - (\overrightarrow{\triangledown}\cdot \overrightarrow{F}) \overrightarrow{G} + (\overrightarrow{G} \cdot \overrightarrow{\triangledown}) \overrightarrow{F} - (\overrightarrow{F} \cdot \overrightarrow{\triangledown})\]

4. Laplacian Function

  1. \[\overrightarrow{\triangledown^{2}} (f + g) = \overrightarrow{\triangledown^{2}} f + \overrightarrow{\triangledown^{2}} g\]

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

  3. \[\overrightarrow{\triangledown^{2}} (fg) = f \overrightarrow{\triangledown^{2}} g + 2 \overrightarrow{\triangledown} f \cdot g + g \overrightarrow{\triangledown^{2}}\]

5. Degree Two Function

  1. \[\overrightarrow{\triangledown} \cdot (\overrightarrow{\triangledown}\times \overrightarrow{F}) = 0\]

  2. \[\overrightarrow{\triangledown}\times (\overrightarrow{\triangledown}f) = 0\]

  3. \[\overrightarrow{\triangledown} \cdot (\overrightarrow{\triangledown} f\times \overrightarrow{\triangledown} g) = 0\]

  4. \[\overrightarrow{\triangledown} \cdot (f\overrightarrow{\triangledown} g - g\overrightarrow{\triangledown} f) = f\overrightarrow{\triangledown^{2}} g - g\overrightarrow{\triangledown^{2}} f\]

  5. \[\overrightarrow{\triangledown}\times (\overrightarrow{\triangledown}\times \overrightarrow{F}) = \overrightarrow{\triangledown} (\overrightarrow{\triangledown} \cdot \overrightarrow{F}) - \overrightarrow{\triangledown^{2}}\]

FAQ (Frequently Asked Questions)

1. What is the Application of Vector Calculus?

Vector calculus has an important role in several fields. Given below are the vector calculus and applications.

  1. Used in the heat transfer

  2. Sports 

  3. Navigation 

  4. Used in the partial differential equations

  5. Used in the three-dimensional geometry

2. What is the Difference Between the Differential Calculus and the Vector Calculus?

Differential calculus refers to the calculus that is related to the derivatives of the functions. Calculus is generally thought to be the differential calculus and the integral calculus. Differential is just a part about the derivatives, whereas the integral is a part about the integrals and the integration.

The variables that are involved in both the differential and the integral calculus are usually taken as the real or the complex numbers, although the different concepts of vector, vector spaces, etc. does not actually enter in. The vector calculus, on the other hand, is related to the aspects of the vector spaces which you treat by using the differential and/or integral calculus. In particular, there are three types of vector quantities which you can form by using the derivatives that are gradient, divergence, and curl. There are theorems too which relate some particular integrals of these quantities to the other integrals.