2017-Fall-ME501-Homework-10-Solutions.pdf

2017-Fall-ME501-Homework-10-Solutions.pdf - 2017 Fall...

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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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