Divergence and Curl
Divergence and Curl are two important operators used in Vector Calculus. Divergence is a scalar operator which tells about the behaviour of a function towards or away from a point. Curl is a vector operator which tells about the behaviour of a function around a point. The vector operator is represented by β which accounts for the partial differentiation of the vector field. The Vector Differential Operator (β) also called Nabla is expressed as β = β/βx i + β/βy j + β/βz k.
Divergence of Vector
If a vector field is given by f(x,y,z) = fxi + fyj + fzk then its divergence is given by taking the scalar of the vector operator is given by
div(f) = β.f(x,y,z) = (β/βx i + β/βy j + β/βz k) Β· (fxi + fyj + fzk )
β β.f(x,y,z) = βx/βx + βy/βy + βz/βz.
Curl of Vector
If a vector field is given by f(x,y,z) = fxi + fyj + fzk then its curl is given by taking the vector of the vector operator
β Γ f(x,y,z) = (β/βx i + β/βy j + β/βz k) β¨― (fxi + fyj + fzk )
β β Γ f(x,y,z) = [Tex]\begin{vmatrix} i & j & k \\ \partial /\partial x& \partial /\partial y & \partial /\partial z \\ f_{x} & f_{y} & f_{z} \\ \end{vmatrix} [/Tex]
β β Γ f(x,y,z) = (βz/βy β βy/βz)i + (βx/βz β βz/βx)j + (βy/βx β βx/βy).
Gradient of Scalar
The gradient of a scalar field F is given by grad(F) or β F. It gives the measurement of the rate and direction of a scalar-valued function. In the Cartesian system, the gradient of a scalar-valued function is given by
β F = (β/βx i + β/βy j + β/βz k)F = β/βx i + β/βy j + β/βz k
Vector Calculus in Maths
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.
Table of Content
- What is Vector Calculus?
- Operation in Vector
- Divergence and Curl
- Vector Calculus Formulas
- Vector Calculus Identities
- Vector Calculus Applications
- Solved Examples
- FAQs
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