Mathematical Physics Vol 1

Chapter 2. Vector analysis

48

Let us now form a sum (usually called the integral sum)

n ∑ i = 1

a ( τ i )( t i − t i

I =

(2.62)

− 1 ) ,

where τ i ∈ ( t i

t i ) .

− 1 ,

Definition If there exists a limit value of the sum I , when n grows indefinitely, where for an arbitrary division of the interval ( t 0 , t n ) the largest of the parts ∆ t i tends to zero, then this limit value is called the finite integral (in the Riemannian sense) of the function a

a ( τ i ) ∆ t i = Z

n ∑ i = 1

t B

a d t .

lim max | ∆ t i |→ 0

(2.63)

t A

1 integral, there also exist Stieltjes, Lebesgue and other

Note that in addition to Riemann

R

integrals.

According to the Newton 2 –Leibniz 3 relation Z t B t A a d t = b t B

t A = b ( t B ) − b ( t A ) ,

(2.64)

if b is a primitive function of the function a .

2.2.3 The line integral of a vector function In previously defined integrals the integration area could be interpreted as a straight line or its part. However, a natural generalization for vector functions, similar to scalar functions, is to extend the integration to curved lines, surfaces, and volumes.

Curve orientation Consider a bounded curve in space, given by the vector equation r = r ( t ) , t ∈ [ t A , t B ] .

(2.65)

To orient the curve means to determine which of the two arbitrary points from the curve is the preceding and which one is the following. 1 Bernhard Riemann (1826-1866), famous German mathematician. He has made significant contributions in geometry, analysis, differential equation theory and number theory. 2 Sir Isaac Newton (1642–1727), famous English physicist and mathematician. Introduced differential and integral calculus simultaneously with Leibniz (although independently of one another). He formulated many basic laws and methods of investigating problems in physics, using mathematical analysis. His book Mathematical Principles of Natural Philosophy, 1687 represents a remarkable contribution to classical mechanics. His work is of great importance for both mathematics and physics. 3 Gottfried Wilhelm Leibniz (1646–1716), German mathematician and philosopher. Introduced differential and integral calculus simultaneously with Newton.