Quiz #1 30/30 congratulations 1)AL-AMER, AHMAD ADNAN MOHA 2)AL-AGEELI, AHMAD IBRAHIM 3)AL-GARNI, BANDAR HASSAN S 4)AL-ARJANI, ALI SAEED ABDU 5)AL-BUGMI, TURKI MAHDI SHA 6)AL-BARAK, MUHAMMAD ABDUL 7)AL-MARRI, ALI MUHAMMAD FA 8)AL-HASAN, KHALED MUHAMMAD 9)AL-MANSOUR, ABDUL-RHMAN M 10)MAKKI, EMAD AHMAD MUHAMMA 11)AL-MESHAL, SAMI MUHAMMAD

2) an indefinite integral MATLAB can find 1) a definite integral syms x Int(x^2, x, 0, 1) syms x Int(x^2, x)

Group # 1 Turki al bogmi Turad al hujile Ahmad aquile Emad Makki Groups

Problem 1

9.9 Line Integral Independent of the Path Evaluate.along the curve C between (-3,-3) and (3,3) (-3,-3) (3,3)

Problem 2

9.9 Line Integral Independent of the Path Evaluate.along the curve C between (-3,-3) and (3,3) (-3,-3) (3,3)

Problem 3

9.9 Line Integral Independent of the Path Evaluate.along the curve C between (-3,-3) and (3,3) (-3,-3) (3,3)

The Integral has the same value The integral Is independent of the path (-3,-3) (3,3 ) (-3,-3) (3,3) (-3,- 3) (3,3)

Under what condition the integral is independent of the path is an exact differential

Test for exact differential is an exact differential

Problem 4

Which line integral is dependent of the path A) C) B) Example3 Example4 HW 7

Problem 5

Application (1) Evaluate.along the curve C between (-3,-3) and (4,4) (4,4) (-3,-3) (4,0) (0,-3)

Problem 6

Evaluate Application (2) (4,4) (-3,-3) (4,0) (0,-3).along the curve C between (-3,-3) and (4,4)

Exact differential is an exact differential There exists a function Such that Example

Theorem 9.8 Fundamental Theorem for Line Integral Suppose there exists a function such that ;that is, is an exact differential. Then depends on only the endpoints A and B of the path C and

Application (1) Evaluate.along the curve C between (-3,-3) and (4,4) (4,4) (-3,-3) (4,0) (0,-3)

How to find Method 2 Method 1 Which method ????? Easy step 1

(-1,0) (3,4)

Notation If Is independent of the path between the endpoints A and B, then we write

Theorem 9.10 Test for Path Independence Is independent of the path

How to find

Conservative Vector Fields If the Is independent of the path, then 2) F is said to be a gradient field 3) F is said to be conservative 4) Is a potential function for F

Conservative Vector Fields In a gradient force field F, 1)The work done by the force upon a particle moving from position A to position B is the same for all paths. 2)The work done by a force along a closed path is zero In a conservative field F, 1)The law of conservation of mechanical energy holds. 2)For a particale moving along a path in a conservative field, kinetic energy + potential energy = constant WHY????

Is independent of the path Conservative Vector Fields

Remarks (pp501) A frictional force such as air resistance is neoconservative. Neoconservative forces are dissipative in that their action reduces kinetic energy without a corresponding increase in potential energy. In other words, if the work done depends on the path, then F is neoconservative.

9.9 Homework

(2,0) (-2,0)

9.9 Homework