The electrostatic potential along a uniform resistive wire of length (metres) is modelled

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch Vector spaces, linear maps and the dual: the Riesz theorem

Problem

The electrostatic potential along a uniform resistive wire of length

L

(metres) is modelled, over a short stretch, as a linear function

V(x)=a+bx,\qquad x\in[0,L],

with

a

in volts and

b

in volts per metre. ((a)) Confirm that the set

\mathcal P_1

of such functions is a vector space under the usual addition and scalar multiplication of functions, and give a basis. ((b)) Construct the dual basis

\{\omega^1,\omega^2\}

\mathcal P_1^*

for the basis

\{1,x\}

. ((c)) The average potential over the wire is the linear functional

\bar V(V)=\dfrac{1}{L}\displaystyle\int_0^L V(x)\,\mathrm dx

. Find

c_1,c_2

such that

\bar V=c_1\omega^1+c_2\omega^2

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