Consider the elliptical-cylindrical coordinate system (eta, psi, z), defined by x = a cos h eta cos psi, y = a sin h eta sin psi, z = z,eta GE 0, 0 LE psi LE 2pi, z R. In PS#6 it was shown that this is an orthogonal coordinate system with scale factors h_1 = h_2 = a (cosh^2 eta - cos^2 psi)^{frac{1}{2}}. Determine the dual bases (E1,E2,E3), ( eta, eta psi, z). Show that: f = a 1/a(cosh^2 eat-cos^s psi)^1/2[ f/ eta e1 + f/ psi e2 + f/ z e3, f/ where (e1, e2, e3) denotes the unit coordinate basis.

Question
Alternate coordinate systems
asked 2020-12-30
Consider the elliptical-cylindrical coordinate system (eta, psi, z), defined by \(\displaystyle{x}={a}{\cos{{h}}}\eta{\cos{\psi}},{y}={a}{\sin{{h}}}\eta{\sin{\psi}},{z}={z},\eta{G}{E}{0},{0}{L}{E}\psi{L}{E}{2}\pi,{z}{R}.{I}{n}{P}{S}#{6}\)
it was shown that this is an orthogonal coordinate system with scale factors \(\displaystyle{h}_{{1}}={h}_{{2}}={a}{\left({{\text{cosh}}^{{2}}\ }\eta-{{\cos}^{{2}}\psi}\right)}^{{{\frac{{{1}}}{{{2}}}}}}.\)
Determine the dual bases \(\displaystyle{\left({E}{1},{E}{2},{E}{3}\right)},{\left(\eta,\eta\psi,{z}\right)}.{S}{h}{o}{w}{t}\hat{:}{f}={a}\frac{{1}}{{a}}\frac{{\left({{\text{cosh}}^{{2}}{e}}{a}{t}-{{\cos}^{{s}}\psi}\right)}^{{1}}}{{2}}{\left[\frac{{f}}{\eta}{e}{1}+\frac{{f}}{\psi}{e}{2}+\frac{{f}}{{z}}{e}{3},\frac{{f}}{{w}}{h}{e}{r}{e}{\left({e}{1},{e}{2},{e}{3}\right)}\right.}\) denotes the unit coordinate basis.

Answers (1)

2020-12-31
Given that magnitude of gradient along x and y direction is \(\displaystyle{h}_{{1}}={h}_{{2}}={a}{\left({{\text{cosh}}^{{2}}\ }\eta-{{\cos}^{{2}}\psi}\right)}^{{{\frac{{{1}}}{{{2}}}}}}.\) and along z direction = h3 =1 Hence delf = vectors in direction of del/magnitude \(\displaystyle={\frac{{{1}}}{{{\left({\text{cosh}{\eta}}-{{\cos}^{{2}}\psi}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}}\rbrace{\left[{\frac{{{\frac{{\partial{f}}}{{\partial\eta}}}}}{{{e}{1}}}}+{\frac{{{\frac{{\partial{f}}}{{\partial\backslash\psi}}}}}{{{e}{2}}}}\right]}+{e}{3}{\frac{{\partial{f}}}{{\partial{z}}}}\)
0

Relevant Questions

asked 2020-11-01
Give a full correct answer for given question 1- Let W be the set of all polynomials \(\displaystyle{a}+{b}{t}+{c}{t}^{{2}}\in{P}_{{{2}}}\) such that \(\displaystyle{a}+{b}+{c}={0}\) Show that W is a subspace of \(\displaystyle{P}_{{{2}}},\) find a basis for W, and then find dim(W) 2 - Find two different bases of \(\displaystyle{R}^{{{2}}}\) so that the coordinates of \(\displaystyle{b}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{5}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\) are both (2,1) in the coordinate system defined by these two bases
asked 2020-11-10
Consider the following linear transformation T : P_2 \rightarrow P_3, given by T(f) = 3x^2 f'. That is, take the first derivative and then multiply by 3x^2 (a) Find the matrix for T with respect to the standard bases of P_n: that is, find [T]_{\epsilon}^{\epsilon}, where- \epsilon = {1, x, x^2 , x^n) (b) Find N(T) and R(T). You can either work with polynomials or with their coordinate vectors with respect to the standard basis. Write the answers as spans of polynomials. (c) Find the the matrix for T with respect to the alternate bases: [T]_A^B where A = {x - 1, x, x^2 + 1}, B = {x^3, x, x^2, 1}.
asked 2020-10-18
Given the elow bases for R^2 and the point at the specified coordinate in the standard basis as below, (40 points) \(\displaystyle{B}{1}=\le{f}{t}{\left\lbrace{\left({1},{0}\right)},{\left({0},{1}\right)}{r}{i}{g}{h}{t}\right\rbrace}&{M}{S}{K}{B}{2}={\left({1},{2}\right)},{\left({2},-{1}\right)}{r}{i}{g}{h}{t}\rbrace{\left({1},{7}\right)}={3}^{\cdot}{\left({1},{2}\right)}-{\left({2},{1}\right)}{N}{S}{K}{B}{2}={\left({1},{1}\right)},{\left(-{1},{1}\right)}{\left({3},{7}={5}^{\cdot}{\left({1},{1}\right)}+{2}^{\cdot}{\left(-{1},{1}\right)}{N}{S}{K}{B}{2}={\left({1},{2}\right)},{\left({2},{1}\right)}{\left({0},{3}\right)}={2}^{\cdot}{\left({1},{2}\right)}-{2}^{\cdot}{\left({2},{1}\right)}{N}{S}{K}{\left({8},{10}\right)}={4}^{\cdot}{\left({1},{2}\right)}+{2}^{\cdot}{\left({2},{1}\right)}{N}{S}{K}{B}{2}={\left({1},{2}\right)},{\left(-{2},{1}\right)}{\left({0},{5}\right)}={N}{S}{K}{\left({1},{7}\right)}=\right.}\) a. Use graph technique to find the coordinate in the second basis. (10 points) b. Show that each basis is orthogonal. (5 points) c. Determine if each basis is normal. (5 points) d. Find the transition matrix from the standard basis to the alternate basis. (15 points)
asked 2020-12-30
Since we will be using various bases and the coordinate systems they define, let's review how we translate between coordinate systems. a. Suppose that we have a basis\(\displaystyle{B}={\left\lbrace{v}_{{1}},{v}_{{2}},\ldots,{v}_{{m}}\right\rbrace}{f}{\quad\text{or}\quad}{R}^{{m}}\). Explain what we mean by the representation {x}g of a vector x in the coordinate system defined by B. b. If we are given the representation \(\displaystyle{\left\lbrace{x}\right\rbrace}_{{B}},\) how can we recover the vector x? c. If we are given the vector x, how can we find \(\displaystyle{\left\lbrace{x}\right\rbrace}_{{B}}\)? d. Suppose that BE is a basis for R^2. If {x}_B = \begin{bmatrix}1 \\ -2 \end{bmatrix}ZSK find the vector x. e. If \(\displaystyle{x}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash-{4}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{f}\in{d}{\left\lbrace{x}\right\rbrace}_{{B}}\)
asked 2021-01-19
Consider the following vectors in \(\displaystyle{R}^{{4}}:\) \(\displaystyle{v}_{{1}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{v}_{{2}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{v}_{{3}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{0}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{v}_{{4}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{0}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\) a. Explain why \(\displaystyle{B}=\le{f}{t}{\left\lbrace{v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}{r}{i}{g}{h}{t}\right\rbrace}\)
forms a basis for \(\displaystyle{R}^{{4}}.\) b. Explain how to convert \(\displaystyle\le{f}{t}{\left\lbrace{x}{r}{i}{g}{h}{t}\right\rbrace}_{{B}},\) the representation of a vector x in the coordinates defined by B, into x, its representation in the standard coordinate system. c. Explain how to convert the vector x into,\(\displaystyle{\left\lbrace{x}\right\rbrace}_{{B}},\) its representation in the coordinate system defined by B
asked 2020-10-20
Consider the linear transformation \(\displaystyle{U}:{R}^{{3}}\rightarrow{R}^{{3}}\) defined by \(\displaystyle{U}{\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{x}\backslash{y}\backslash{z}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{z}-{y}\backslash{z}+{y}\backslash{3}{z}-{x}-{y}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}\) and the bases PSK\epsilon = \left\{ \left(\begin{array}{c}1\\ 0 \\0\end{array}\right), \left(\begin{array}{c}0\\ 1 \\ 0\end{array}\right), \left(\begin{array}{c}0\\ 0 \\ 1\end{array}\right) \right\}, \gamma = \left\{ \left(\begin{array}{c}1 - i\\ 1 + i \\ 1 \end{array}\right), \left(\begin{array}{c} -1\\ 1 \\ 0\end{array}\right), \left(\begin{array}{c}0\\ 0 \\ 1\end{array}\right) \right\}, Compute the four coordinate matrices \(\displaystyle{{\left[{U}\right]}_{{\epsilon}}^{{\gamma}}},{{\left[{U}\right]}_{{\gamma}}^{{\gamma}}},\)
asked 2020-11-05
Consider the linear transformation \(\displaystyle{U}:{R}^{{3}}\rightarrow{R}^{{3}}\) defined by \(\displaystyle{U}{\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{x}\backslash{y}\backslash{z}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{z}-{y}\backslash{z}+{y}\backslash{3}{z}-{x}-{y}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}\) and the bases PSK\epsilon = \left\{ \left(\begin{array}{c}1\\ 0 \\0\end{array}\right), \left(\begin{array}{c}0\\ 1 \\ 0\end{array}\right), \left(\begin{array}{c}0\\ 0 \\ 1\end{array}\right) \right\}, \gamma = \left\{ \left(\begin{array}{c}1 - i\\ 1 + i \\ 1 \end{array}\right), \left(\begin{array}{c} -1\\ 1 \\ 0\end{array}\right), \left(\begin{array}{c}0\\ 0 \\ 1\end{array}\right) \right\}, Compute the four coordinate matrices \(\displaystyle{{\left[{U}\right]}_{{\epsilon}}^{{\epsilon}}},{{\left[{U}\right]}_{{\gamma}}^{{\epsilon}}}\)
asked 2021-02-25
Consider the bases \(\displaystyle{B}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash{5}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}{o}{f}{R}^{{2}}{\quad\text{and}\quad}{C}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}{o}{f}{R}^{{3}}\).
and the linear maps PSKS \in L (R^2, R^3) and T \in L(R^3, R^2) given given (with respect to the standard bases) by \(\displaystyle{\left[{S}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}&-{1}\backslash{5}&-{3}\backslash-{3}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{\quad\text{and}\quad}{\left[{T}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&-{1}&{1}\backslash{1}&{1}&-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\) Find each of the following coordinate representations. \(\displaystyle{\left({a}\right)}{\left[{S}\right]}_{{{B},{E}}}\)
asked 2021-02-21
All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V. V is \(\displaystyle{R}^{{2}},{S}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},{v}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash-{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\\)
asked 2020-11-14
Consider the bases \(\displaystyle{B}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash{5}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}{o}{f}{R}^{{2}}{\quad\text{and}\quad}{C}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}{o}{f}{R}^{{3}}\).
and the linear maps PSKS \in L (R^2, R^3) and T \in L(R^3, R^2) given given (with respect to the standard bases) by \(\displaystyle{\left[{S}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}&-{1}\backslash{5}&-{3}\backslash-{3}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{\quad\text{and}\quad}{\left[{T}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&-{1}&{1}\backslash{1}&{1}&-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\) Find each of the following coordinate representations. \(\displaystyle{\left({b}\right)}{\left[{S}\right]}_{{{E},{C}}}\)
\(\displaystyle{\left({c}\right)}{\left[{S}\right]}_{{{B},{C}}}\)
...