Vector Differential Calculus

Contents

Introduction

Gradient of Scalar Point Function

Directional Derivative

Divergence & Curl of Vector Point Function

Scalar Potential

1 M Questions

Introduction

In this chapter vector differential calculus is considered, which extends the basic concepts of (ordinary) differential, such as continuity and differentiability to vector function in simple and natural way. Also, the new concepts of gradient, divergence and curl are introduced.

Scalar Point Function

If to each point \( P(x,y,z) \) of a region \( R \) in space, a unique scalar, denoted by \( \phi (x,y,z) \) is associated, then \( \phi (x,y,z) \) is called scalar point function. The region \( R \) so define is called a scalar field.

Examples:
  1. \( P = x^{2} + y^{2} \) is a scalar point function for each point \( P(x,y) \) and it forms a two-dimensional scalar field.
  2. \( P\left( x,y,z \right) = x^{2} + y^{2} + z^{2} \) is a scalar point function for each point \( P(x,y,z) \) and it forms a three-dimensional scalar field.
  3. Physical examples of a scalar field are:
    1. The mass density of the atmosphere.
    2. The temperature at each point in an insulated wall.
    3. The water pressure at each point in an ocean.

Vector Point Function

If to each point \( P(x,y,z) \) of a region \( R \) in space a unique vector, denoted by \( \vec{F} \) is associated, then \( \vec{F}\left( x,y,z \right) \) is called a vector point function. The region \( R \) so defined is called a vector field.

Examples:
  1. \( \bar{F} = xi + yj + zk \) is a vector point function, which associates with each point \( \left( x,y,z \right) \) a vector pointing away from the origin. This represents a three-dimensional source field.
  2. The velocity \( \vec{v} \) of a particle at any point P of a fluid occupying a region is a vector point function.

Level Surface

Let \( \phi (x,y,z) \) be a non-zero scalar point function and \( c \) be any real constant. The set of points \( P \) such that \( \phi (P) = c \) is called level surface.

Example:

Equipotential functions and isothermal functions are examples of level surfaces.

The Vector Differential Operator

The vector differential operator \( \nabla \) (read as 'del') is defined as

\[ \nabla = \hat{i}\frac{\partial }{\partial x} + \hat{j}\frac{\partial }{\partial y} + \hat{k}\frac{\partial }{\partial z} \]

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