See, however, the unpublished text Applications of Classical Physics, Vol. ε ijk is a totally anti-symmetric tensor defined by: ε ijk = 1 if ijk is an even permutation of 123, ε ijk = - 1 if ijk is an odd permutation of 123, and ε ijk = 0 otherwise. We consistently use superscripts for coordinate indices and subscripts for component indices of vectors and tensors these have nothing to do with covariant and contravariant components. No appropriate reference could be found in the American Journal of Physics, the European Journal of Physics, the Physics Teacher, or using broader searches. The shortness of the reference list for this paper underscores the absence of shear and gradient of a vector in undergraduate physics.
Weber, Mathematical Methods for Physicists, 6th ed. The gradient ∇ A of a vector field A, on the other hand, describes the local variation of the vector field as one moves from point to point. Helmholtz’s theorem is a uniqueness theorem about a global solution to a differential equation-namely, if we know the divergence and curl of a vector field everywhere in space, then we can integrate these expressions to find the field. 4) which states that any differentiable vector field is uniquely specified by giving its divergence and curl within a simply connected region of space and its normal component on the boundary.
The reader should note that our statement about the necessity of knowing the gradient of a vector field does not conflict with Helmholtz’s theorem (see e.g., Sec. Griffiths, Introduction to Electrodynamics, 3rd ed. Purcell, Electricity and Magnetism, Berkeley Physics Course, Vol. Geometric explanations of divergence and curl are also presented in the excellent texts by E. Schey, div, grad, curl, and all that, an informal text on vector calculus, 3rd ed.
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