The surface integral of V = x + yŷ + (z - 3)2 over the sphere of unit radius centred at the origin is Select one: ○ a. 4π ○ b. O ○ c. 4π/3 ○ d. π

Given the vectors i =(- 3, 7, 4] and y = [- 6, 0, 8], determine b) proj „ . a) The angle between i and ỹ

(b) Sphericals x=r sine coso y=r sine sino z=r cose Consider spherical coordinates as defined by their relationships to the Cartesian coordinates: x =r sin 8 cos p X2 =r sin 0 cos X3 =r cos 0 r2 = [xỉ + x3 + x e = tan-(x? + x3)x3 $ = tan-(x2/x1) or

If A = 2yz i – (x + 3y – 2)j+ (x² + z)k, evaluate SS, (V × A)· dS over the surface of intersection of the cylinders x² +y? = 1, x? + z² = 1 which is included in the first octant.

A circular area S is concentric with the origin, has radius a, and lies in the yz-plane. Calculate ∫??⃗·?ˆ??∫SE→·n^dA for ?⃗=3?2?ˆ.E→=3z2i^.

40 Q.14 The integral of the vector Ä(p, q, z) coso ô (standard notation for cylindrical coordinates is used) over the volume of a cylinder of height L and radius R, is: (A) 207R,L (î + j) (B) 0 (C) 40πR, Lj ( D) 40πR, L t

1. Two points in the æy plane have Cartesian coordinates (2.00, –4.00) m and (-3.00, 3.00) m. Determine (a) the distance between these points and Answer + (b) their polar coordinates. Answer

518۹ من ٥۱۹۰ Example 2; find O between A¯and B? A*=2î +3ƒ +k , B =3î -2j +k Example 3; if 0=30o between A¯and B'find q value? A¯=2î +2j +4k , B'=qî +j -2k Example 4; Find the angle between the vectors A¯=2î +3ƒ +1k , B` =-4î +2ƒ -1k منو يعرف يحل ذني الاسئله مالت الفيزياء de between4md 8 agnitude 2 and vect he angle a between 90, or 180. following situations must be. On each sit an one correct answ C , (c) A 8-6 B=30 betweenA"an d the angle between t N--4i +2 -1 دعاء صاحب حسین 4:19p äclul 21.02.04 II

Using spherical polar coordinates, find Vf for (a) f = r4 (b) f = 2+1 (c) f = r² sin 20 cos o

Set 3: Find the midpoint between each pair of points. 1. (0, 8) and (8, -6) C. (8, -9) and (8, 5) М 19. (3, 7) and (-9, 5) R. (5, -6) and (-1, -8) M = 10. (6, 3) and (10, -7) M = L. (4, -1) and (6, 1) M 3= 4. (9, -4) and (-5, -10) T. (-1, -7) and (9, 9) M = M = 12. (8, -3) and (2, 3) w. (-3, 3) and (-3, 9) Set 4: Find the midpoint between each pair of points.