By Firk F.W.K.

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1-2 Calculate the second partial derivatives of f(x, y) = (1/√y)exp{–(x – a)2/4y}, a = constant. 1-3 Check the answers obtained in problem 1-2 by showing that the function f(x, y) in 1-2 is a solution of the partial differential equation ∂2f/∂x2 – ∂f/∂y = 0. 1-4 If f(x, y, z) = 1/(x2 + y 2 + z 2)1/2 = 1/r, show that f(x, y, z) = 1/r is a solution of Laplace’s equation ∂2f/∂x2 + ∂2f/∂y2 + ∂2f/∂z2 = 0. This important equation occurs in many branches of Physics. 1-5 At a given instant, the radius of a cylinder is r(t) = 4cm and its height is h(t) = 10cm.

This coordinate transformation has “diagonalized” the original matrix: –2 1 1 –2 → –1 0 0 –3 . The matrix with zeros everywhere, except along the main diagonal, has the interesting property that it does not rotate the vectors on which it acts. The diagonal elements are called the eigenvalues of the diagonal matrix, and the renormalized vectors are called eigenvectors. A small industry exists that is devoted to finding optimum ways of diagonalizing large matrices. Illustrate the motion of the system in the x-y frame and in the u-v frame.

Uk can be written linearly in terms of the remaining ones we say that the vectors are linearly independent. ckuk = 0 , in which the scalars ci are not all zero. 0] ... xnen. 38) The ei’s are said to span the space of all n-vectors; they form a basis. Every basis of an nspace has exactly n elements. The connection between a vector x and a definite coordinate system is made by choosing a set of basis vectors ei. 10 The geometry of vectors The laws of vector algebra can be interpreted geometrically for vectors of dimension 2 and 3.