Unformatted text preview: 2017 Fall Semester ME 501 Mechanical Engineering Analysis Homework 10: Vector calculus I Solutions: 1. Assume that three vertexes of a triangle are given by the position vectors in some ,
1,2,3. Find the area of this triangle in Cartesian coordinates, terms of coordinates of its vertexes, , , … (Use cross product). Solution: 1
 det
2  2. In Cartesian coordinates, some plane contains point 1,2, 1 and vectors 3 and 2
3 . Find equation of this plane and coordinate of a point on this plane with 10 and 10 (Use triple product). Solution: 9 6 5 35/2 3. Find the volume of tetrahedron with vertexes in points (1,3,6), (3,7,12), (8,8,9), and (2,2,8). Solution: 2
1
det 7
6
1 1
det
6
4. A curve is given parametrically by the equation to this curve in point P with coordinates (2,0.5,0) . 4
5
1
1/ 6
3
2 15 . Find tangent vector Solution: 1/ 2 2
0.25 5. Find the length of a circular helix given by the equation from point 4,0,0 to point 4,0,10 . Solution: 2 √41 4 cos 4 sin 5 cos 6. Find total length of the hypocycloid curve given by the equation sin .   3 cos sin  /   4 6 . 7. Find a Consider surface given by the equation 6
2
225. Represent the surface equation in the parametric form first), Find equations for the normal vector and then find components of this vector in the point with coordinates (5,5,5). Solution: 6
225
u, v 15
u, v u, v 1
√6 2
225 1 1
√6 √2 sin sin
1 cos cos √6
1
225
sin
√2 In point (5,5,5), cos 1 sin cos 15
15 1
225 1 sin sin 1
√2 , cos 100 , 8. Force field is given by the equation work of this work along the trajectory 2 cos 2,0,0 to point 2,2 , 0 . Is this work path‐independent? Solution: ∙ 2 8 sin cos cos sin √2
1
sin sin
√6 cos cos sin
1
2√3 sin cos √ √ . Calculate the 2 sin from point 9. Find gradient of the scalar field , / . Solution: 10.The temperature field in a medium with constant thermal conductivity is given by the . Calculate the components of heat flux in the medium. . scalar field /
Solution: 2
2
11. , , and , , are given scalar and vector fields. Prove that div div ∙ (Write LHS and RHS in terms of individual derivatives and compare them with each other). Solution: LHS = RHS if written in the component form 12. , ,
2
2
4 and , ,
3
vector fields. Calculate value of curl ∙ in the point (4,0,2). are two Solution: curl ∙
192 13.In a fluid flow, the velocity potential is given by the equation ln
. Find the components of the fluid velocity in Cartesian coordinates. Is this flow incompressible? Irrotational? Why? Solution: 2 2 Flow irrotational because curl 0 for any gradient field . Flow is incompressible becausediv 0. 14.Assume that the fluid flow has the velocity field a. Is this flow irrotational? Why? Solution: Flow is irrotational becausecurl 0. b. Is this flow incompressible? Why? Solution: Flow is compressible becausediv 0. . ...
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 Spring '14
 WillardC.Schreiber

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