Engineering Mathematics: Line Integrals and Vector Calculus, Lecture notes of Engineering Mathematics

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MAT 247 Engineering Mathematics

Hakkı Ula¸s Unal¨ Dept. of Electrical-Electronics Eng. Eskisehir Technical University, Turkey

October 10, 2018

Today

(^1) Vector Calculus

Line Integral

(^0 2 4) Hours 6 8 10

Speed

3

4

5 Speed profile

(^00 2 4 6 8 )

1

2

3

4

5

How far you are from the reference

Distance

∫ (^) t= t=0 f^ (τ^ )dτ^ is the integral of function^ f^ (t) from^ t^ = 0 to^ t^ = 10.

Line Integral

Mass of a wire

Total mass of a wire, where the mass of a per unit length is described as.

1 1. 0 0. (^20) -1 -0.

4

6

-0.

0

1

8

Some notation and preliminaries

In general, we represent a curve C by a parametric representation

r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k, a ≤ t ≤ b

C is called as a closed curve, if its initial point coincides with its terminal point

Some notation and preliminaries

In general, we represent a curve C by a parametric representation

r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k, a ≤ t ≤ b

C is called as a closed curve, if its initial point coincides with its terminal point C called a smooth curve if it has a unique tangent, whose direction varies cts throughout the curve, at each point of C (in other words, it can be differentiable sufficiently many times)

Some general rules

C

kF(r) · dr = k

C

F(r) · dr ∫

C

(F(r) + G(r)) · dr =

C

F(r) · dr +

C

G(r) · dr ∫

C

F(r) · dr =

C 1

F(r) · dr +

C 2

F(r) · dr

Invariance of line integral wrt representation

Let r(t), a ≤ t ≤ b, and q(ν), γ ≤ ν ≤ β, be representations of a curve C. Then, (^) ∫

C

F(r) · dr =

C

F(q) · dq,

Line Integral

Line Integral

A line integral of a vector function F(r) over a curve C described as r(t) is defined by (^) ∫

C

F(r) · dr,

since r(t) = [x(t), y(t), z(t)], dr = [dx(t), dy(t), dz(t)], and F(·) = [F 1 , F 2 , F 3 , ] ∫

C

F(r) · dr =

C

(F 1 dx + F 2 dy + F 3 dz).

Then, since x′^ = dx/dt and y′^ = dy/dt and z′^ = dz/dt, where a ≤ t ≤ b. ∫

C

F(r) · dr =

∫ (^) b

a

F 1 x′^ + F 2 y′^ + F 3 z′

dt.

Some applications of line integral

20 0 10

A

B

0

10

20

0

5

10

15

20

Work done by Force

In some cases, we should know how much electromagnetic force should be provided to move an object from A to B (seen in the figure).

W =

C

F(r) · dr,

Path dependence

W =

C

F(r) · dr,

does not only depend on F, also it depends on your path.

C

F(r) · dr =

0

[−t, −(1 − t)] · [1, −1]dt = 0,

A simple Example

Evaluate the integral of F(r(t)) = [−y(t), −x(t)] over r(t) = [(1 − t), 0] + [0, t], where 0 ≤ t ≤ 1.

y

A^ x

B

C

F(r) · dr,

does not depend on how you reach to B from A in a defined domain.

A

B

C

F(r) · dr,

with some continuous F 1 , F 2 , F 3 in domain D is path independent iff

F = gradf,

holds for some f in D.