8/11/2019 p set on MMOP

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PH 205: Mathematical methods of physics

Problem Set 1

1. A simple LC circuit consists of an inductor of inductance L connected to a capacitor of capacitance C.

(a) What is the differential equation obeyed by I(t), the current in the circuit as a function of time t?(b) Show that all possible solutions of the above differential equation form a vector space over R. What is the

dimension of this vector space?

(c) What are appropriate orthonormal basis vectors for this vector space (i.e. the inner product of the basisvectors is zero if they are different and equal to unity if they are the same)? What is a suitable definitionof this inner product?

2. Now suppose there are two identical LC circuits of the sort described in the previous problem but with theirinductors coupled through a mutual inductance M. Assume that the self inductanceL and capacitance C ofeach circuit are the same as in the previous problem. The currents in the two circuits as functions of time arerespectivelyI1(t) and I2(t).

(a) What are the differential equations for I1(t) and I2(t)?

(b) Show that all possible solutions I1(t) and I2(t) form a vector space over R. What is the dimension of thisspace?

(c) Assume thatM= 3L/5. In analogy with 1(c), define appropriate orthonormal basis vectors with a suitableinner product.

(d) Will the choice of inner product for the previous part work for any value ofM?

3. Now assume that we write the currents in problem 2 as I(t) = Re[I(t)], where I(t) =

I1(t)I2(t)

is an array of

complex currents I1(t) and I2(t) for the two circuits.

(a) Argue that I(t) forms a vector space over C, the field of complex numbers.

(b) Now, show that a vector Ican be defined corresponding to I(t) in terms of basis vectors ei which is givenby

I=iiei,

whereei C. What are the appropriate ei and what is the minimum number required?

(c) What is a suitable definition of the scalar product ei.ej in analogy with 2(c) for M= 3L/5?

4. A set of linearly independent vectors {ui} spans a vector space Vover a field F.

(a) Show that this is the largest possible linearly independent set of vectors in V.

(b) Show that any smaller linearly independent set of vectors does not span V.

(c) As a consequence of the above two results, every set of linearly independent vectors which spans Vhas thesame size, which is the dimension ofV, dim(V).

5. In this problem you will prove certain statements relating to subspaces of a vector space.

(a) IfU andWare both subspaces ofVover a field F, show that UW is also a subspace ofV over F.(b) The sum space U+ W is the set of all vectors in Vwhich can be written as v= u +w, where u U andw W. Show that U+ Wis also a vector space over F.

(c) Finally show that dim(U+ W) = dim(U)+dim(W) dim(UW), where dim denotes the dimension of avector space.

6. A vectors space V is a direct sum of two vector spaces U and W (denoted by V =U W) ifU W = . Allthree spaces V, U and W are defined over the same field F. Let V = R3 and F = R. Is V = UW in thefollowing two cases?

(a) U is the xy plane and W is the yz plane.

(b) U is the xy plane and W is the z axis.