References
49
2.11 Unit base vectors for any coordinate system can be computed using the
equation
e i =
r ,i
|r ,i |
,
in which r is the position vector from the origin to any point in space, and
comma denotes differentiation with respect to coordinate x i . Clearly, this
relation holds in Cartesian coordinates, where
r = xe x + ye y + ze z .
(a) Cartesian and spherical coordinates are related by
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ.
Determine the unit base vectors for spherical coordinates in terms of e x ,
e y , and e z .
(b) Using the results from part (a), write the differential vector dr in the
form
dr = ds r e r + ds θ e θ + ds φ e φ .
(c) Write the gradient operator ∇ and the Laplacian ∇ 2 in spherical
coordinates.
2.12 If a and b are vector functions and T a second-order tensor function of
position, show the following:
(a) ∇(T · a) = (∇T) · a + (∇a) · T
T
(b) ∇ · (a × b) = (∇ × a) · b − a · (∇ × b)
Hint: Work in Cartesian coordinates.
References
Drew TB (1961) Handbook of vector and polyadic analysis. Reinhold Publishing Corporation,
New York
Einstein A (1916) The foundation of the general theory of relativity. Annalen Phys 14:769–822
Flugge W (1972) Tensor analysis and continuum mechanics. Springer, New York
Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley,
New York
Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall,
Englewood Cliffs
Simmonds JG (1994) A brief on tensor analysis. Springer, New York
49
2.11 Unit base vectors for any coordinate system can be computed using the
equation
e i =
r ,i
|r ,i |
,
in which r is the position vector from the origin to any point in space, and
comma denotes differentiation with respect to coordinate x i . Clearly, this
relation holds in Cartesian coordinates, where
r = xe x + ye y + ze z .
(a) Cartesian and spherical coordinates are related by
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ.
Determine the unit base vectors for spherical coordinates in terms of e x ,
e y , and e z .
(b) Using the results from part (a), write the differential vector dr in the
form
dr = ds r e r + ds θ e θ + ds φ e φ .
(c) Write the gradient operator ∇ and the Laplacian ∇ 2 in spherical
coordinates.
2.12 If a and b are vector functions and T a second-order tensor function of
position, show the following:
(a) ∇(T · a) = (∇T) · a + (∇a) · T
T
(b) ∇ · (a × b) = (∇ × a) · b − a · (∇ × b)
Hint: Work in Cartesian coordinates.
References
Drew TB (1961) Handbook of vector and polyadic analysis. Reinhold Publishing Corporation,
New York
Einstein A (1916) The foundation of the general theory of relativity. Annalen Phys 14:769–822
Flugge W (1972) Tensor analysis and continuum mechanics. Springer, New York
Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley,
New York
Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall,
Englewood Cliffs
Simmonds JG (1994) A brief on tensor analysis. Springer, New York