8-13. A force field F is said to be conservative if it is derivable from a scalar
potential function V such that
F=±grad V.
One uses either a plus sign or a minus sign depending upon the particular application
being represented.
Consider the motion of a spring-mass system which oscillates in the x-direction.
Assume the force acting on the mass m is derivable from the potential function
V =^12 kx^2 ,where k is the spring constant. Use Newton’s second law (vector form)
and derive the equation of motion of the spring-mass system.
8-14. (Divergence of a vector quantity )
Let
F(x, y, z) = F 1 (x, y, z )ˆe 1 +F 2 (x, y, z)ˆe 2 +F 3 (x, y, z)ˆe 3
denote a vector field and consider a volume element ∆x∆y∆z located at the point
(x, y, z )in this vector field.
(a) Use the first couple of terms of a Taylor series expansion to calculate the vector
field at
(i) F(x+ ∆ x, y, z )
(ii) F(x, y + ∆ y, z)
(iii) F(x, y, z + ∆ z)
(b) Use the results in part (a) and calculate the flux over the surface of the cubic
volume element ∆V = ∆ x∆y∆zand then divided by the volume of this element
in the limit as the volume tends toward zero.
8-15. Determine whether the given vector fields are solenoidal or irrotational
(i) F = (2xyz −z^2 )eˆ 1 +x^2 zeˆ 2 + (x^2 y− 2 xz)ˆe 3
(ii) F =eˆ 1 + (x^2 y−y^2 z)eˆ 2 + (yz^2 −x^2 z)ˆe 3
(iii) F = 2xy ˆe 1 + (x^2 − 2 yz )ˆe 2 −y^2 ˆe 3
(iv) F = 2x(z−y)ˆe 1 + (y^2 −yx^2 )ˆe 2 + (zx^2 −z^2 )ˆe 3
8-16. Show that div (curl F) = 0
8-17. Show that curl (grad φ) = 0
