A particle exists in three dimensions and is trapped inside a solid S. The cross section of the cylinder C on the xy plane is the region bounded between r = cos (0) and r = sin (0) in the first quadrant. All the points in the solid S exists inside the cylinder C bounded between the planes z = -y and z = y. Before we can get into the particle dynamics, we need to introduce the notion of "inner product." You are already familiar with the dot product of two vectors. Dot products give a sense of how much the two vectors are aligned with each other. In layman's terms, a function can be thought of as a vector with infinite components. Lets put this in perspective, a 3d vector ā = api + ayj +a;k has three components a, ay, and az. Now, lets consider a function f defined by the map ƒ : x → 7x, where x E [-3, 2] ; then the "xth" component of the "vector" f is just f (x), or equivalently 7x. Since x can take the value of any real number between –3 and 2 and that there are infinite real numbers between any two numbers, then it follows that the "vector" f has infinite components. In the case of the dot product, we have to take a summation, and in the case of an inner product, we have to take an integral. The definition of the inner product of two functions g and h (denoted by (9|h)) over a volume S is defined as g (x, y, z)* h (x, y, z) dV, where g (x, y, 2)* is the complex conjugation of g (x, y, 2). The probability of finding the particle anywhere outside the solid S is 0 and the probability of finding the particle inside the solid S is non zero. The probability density function (PDF in short) which describes this fact mathematically is PDF = v (x, y, z)* Þ (x, Y, z) , (8) where v is (x, y, z) lies within S (7²+y²){ otherwise, 0, b (x, y, z) = where A is a positive constant. (a) Write down the value of (|4), explain your reasoning in one sentence. (Your answer should be a number, do not evaluate any integral) [2] (b) Now write down the integral(s) with the limits in order to evaluate (|) using equation (7) and explain why cylindrical coordinates will be a good choice to evaluate the integral (Do not evaluate the integral yet)

A particle exists in three dimensions and is trapped inside a solid S. The cross section of the cylinder C on the xy plane is the region bounded between r = cos (0) and r = sin (0) in the first quadrant. All the points in the solid S exists inside the cylinder C bounded between the planes z = -y and z = y. Before we can get into the particle dynamics, we need to introduce the notion of "inner product." You are already familiar with the dot product of two vectors. Dot products give a sense of how much the two vectors are aligned with each other. In layman's terms, a function can be thought of as a vector with infinite components. Lets put this in perspective, a 3d vector ā = api + ayj +a;k has three components a, ay, and az. Now, lets consider a function f defined by the map ƒ : x → 7x, where x E [-3, 2] ; then the "xth" component of the "vector" f is just f (x), or equivalently 7x. Since x can take the value of any real number between –3 and 2 and that there are infinite real numbers between any two numbers, then it follows that the "vector" f has infinite components. In the case of the dot product, we have to take a summation, and in the case of an inner product, we have to take an integral. The definition of the inner product of two functions g and h (denoted by (9|h)) over a volume S is defined as g (x, y, z)* h (x, y, z) dV, where g (x, y, 2)* is the complex conjugation of g (x, y, 2). The probability of finding the particle anywhere outside the solid S is 0 and the probability of finding the particle inside the solid S is non zero. The probability density function (PDF in short) which describes this fact mathematically is PDF = v (x, y, z)* Þ (x, Y, z) , (8) where v is (x, y, z) lies within S (7²+y²){ otherwise, 0, b (x, y, z) = where A is a positive constant. (a) Write down the value of (|4), explain your reasoning in one sentence. (Your answer should be a number, do not evaluate any integral) [2] (b) Now write down the integral(s) with the limits in order to evaluate (|) using equation (7) and explain why cylindrical coordinates will be a good choice to evaluate the integral (Do not evaluate the integral yet)

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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A particle exists in three dimensions and is trapped inside a solid S. The
cross section of the cylinder C on the xy plane is the region bounded between
r = cos (0) and r = sin (0) in the first quadrant. All the points in the solid S
exists inside the cylinder C bounded between the planes z = -y and z = y.
Before we can get into the particle dynamics, we need to introduce the
notion of "inner product." You are already familiar with the dot product of
two vectors. Dot products give a sense of how much the two vectors are
aligned with each other. In layman's terms, a function can be thought of as a
vector with infinite components. Lets put this in perspective, a 3d vector
a = a,i + a,j + a,k has three components a, ay, and az. Now, lets consider a
function f defined by the map f : x+ 7x, where x E -3,2] ; then the "xth"
component of the "vector" f is just f (x), or equivalently 7. Since x can
take the value of any real number between -3 and 2 and that there are
infinite real mumbers between any two numbers, then it follows that the
"vector" f has infinite components. In the case of the dot product, we have
to take a summation, and in the case of an inner product, we have to take an
integral. The definition of the inner product of two functions g and h
(denoted by (9|h)) over a volume S is defined as
(glh) = ||| 9 (7, 4, z)* h (x, y, 2) dV,
(7)
where g (x, y, z)* is the complex conjugation of g (x, y, z).
The probability of finding the particle anywhere outside the solid S is 0 and
the probability of finding the particle inside the solid S is non zero. The
probability density function (PDF in short) which describes this fact
mathematically is
PDF
v (x, y, 2)* » (x, Y, 2),
where y is
(x, y, z) lies within S
V (x, y, z) =
(x2+y2):
0, otherwise,
where A is a positive constant.
(a) Write down the value of (4), explain your reasoning in one sentence.
(Your answer should be a number, do not evaluate any integral)
[2]
(b) Now write down the integral(s) with the limits in order to evaluate (l)
using equation (7) and explain why cylindrical coordinates will be a good
choice to evaluate the integral (Do not evaluate the integral yet)

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