Page 92 - Introduction to Continuum Mechanics

Tensors 77

2C6. Let <t>(xyj) and V(xyj:) be scalar fields, and let v(:t,y,z) and w(jc,y,z) be vector fields. By
writing the subscripted component form, verify the following identities:
(a) V(0+#>) = V0+VV
Sample solution:

WV» "I t-'-'V/
(b) div(v+w) = diw+divw,
(c) div(0v) = (V0)-v+0(diw),
(d) curl(V0) = 0,
(e) div(curlv) = 0.
2 2 2
2C7, Consider the vector field v = x e\ + z 63 + y 63. For the point (1,1,0):
(a) Find the matrix of Vv.
(b) Find the vector (Vv)v.
(c)Finddivv and curlv.

(d) if dr — ds(ei + «2 + e s)»find the differential d\.
2D1. Obtain Eq. (2D1.15)
2D2. Calculate div u for the following vector field in cylindrical coordinates:
2
(a)w r = UQ = 0, u z = A + Br ,
„ . sin^ „ n
(b)w r = -^-, w e = 0, u z = 0,

2D3. Calculate div u for the following vector field in spherical coordinates:

2D4. Calculate Vu for the following vector field in cylindrical coordinate

2D5. Calculate Vu for the following vector field in spherical coordinate