Vector Calculus

VECTOR CALCULUS

Parameterised curves and arc length, tangents and normals to curves in R3; curvature and torsion. [1] Integration in R2 and R3 Line integrals. Surface and volume integrals: deﬁnitions, examples using Cartesian, cylindrical and spherical coordinates; change of variables. [4]

Directional derivatives. The gradient of a real-valued function: deﬁnition; interpretation as normal to level surfaces; examples including the use of cylindrical, spherical ∗and general orthogonal curvilinear∗ coordinates. Divergence, curl and ∇2 in Cartesian coordinates, examples; formulae for these operators (statement only) in cylindrical, spherical ∗and general orthogonal curvilinear∗ coordinates.
Solenoidal ﬁelds, irrotational ﬁelds and conservative ﬁelds; scalar potentials. Vector derivative identities. [5]

Divergence theorem, Green’s theorem, Stokes’s theorem, Green’s second theorem: statements; informal proofs; examples; application to ﬂuid dynamics, and to electromagnetism including statement of Maxwell’s equations. [5]

Laplace’s equation in R2 and R3: uniqueness theorem and maximum principle. Solution of Poisson’s equation by Gauss’s method (for spherical and cylindrical symmetry) and as an integral. [4]

Tensor transformation laws, addition, multiplication, contraction, with emphasis on tensors of second rank. Isotropic second and third rank tensors. Symmetric and antisymmetric tensors. Revision of principal axes and diagonalization. Quotient theorem. Examples including inertia and conductivity. [5]

H. Anton Calculus. Wiley Student Edition 2000 T.M.
Apostol Calculus. Wiley Student Edition 1975
M.L. Boas Mathematical Methods in the Physical Sciences. Wiley 1983()
† D.E. Bourne and P.C. Kendall Vector Analysis and Cartesian Tensors. 3rd edition, Nelson Thornes 1999
E. Kreyszig Advanced Engineering Mathematics. Wiley International Edition 1999(pdf)
J.E. Marsden and A.J.Tromba Vector Calculus. Freeman 1996
P.C. Matthews Vector Calculus. SUMS (Springer Undergraduate Mathematics Series) 1998
† K. F. Riley, M.P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering. Cambridge University Press 2002(pdf)
H.M. Schey Div, grad, curl and all that: an informal text on vector calculus. Norton 1996(pdf)
M.R. Spiegel Schaum’s outline of Vector Analysis. McGraw Hill 1974(pdf)