MathsGee Learning Club - Recent questions tagged coordinate
https://mathsgee.com/tag/coordinate
Powered by Question2AnswerFind the number of solutions to \[ \sec \theta+\csc \theta=\sqrt{15} \] where \(0 \leq \theta \leq 2 \pi\).
https://mathsgee.com/42231/find-the-number-solutions-sec-theta-csc-theta-sqrt-where-theta
Find the number of solutions to<br />
\[<br />
\sec \theta+\csc \theta=\sqrt{15}<br />
\]<br />
where \(0 \leq \theta \leq 2 \pi\).Mathematicshttps://mathsgee.com/42231/find-the-number-solutions-sec-theta-csc-theta-sqrt-where-thetaSun, 10 Jul 2022 07:29:02 +0000Fix \(b>a>0\), define \[ \Phi(r, \theta)=(r \cos \theta, r \sin \theta) \] for \(a \leq r \leq b, 0 \leq \theta \leq 2 \pi\).
https://mathsgee.com/41589/fix-define-theta-cos-theta-sin-theta-for-leq-leq-leq-theta-leq
Fix \(b>a>0\), define<br />
\[<br />
\Phi(r, \theta)=(r \cos \theta, r \sin \theta)<br />
\]<br />
for \(a \leq r \leq b, 0 \leq \theta \leq 2 \pi\). (The range of \(\Phi\) is an annulus in \(R^{2}\).) Put \(\omega=x^{3} d y\), and compute both<br />
\[<br />
\int_{\Phi} d \omega \text { and } \int_{\partial \Phi} \omega<br />
\]<br />
to verify that they are equal.Mathematicshttps://mathsgee.com/41589/fix-define-theta-cos-theta-sin-theta-for-leq-leq-leq-theta-leqTue, 05 Jul 2022 04:58:43 +0000The surface of a two-dimensional ellipsoid can be imbedded in three-dimensional Euclidean space by the equation \[ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \]
https://mathsgee.com/41448/dimensional-ellipsoid-imbedded-dimensional-euclidean-equation
The surface of a two-dimensional ellipsoid can be imbedded in three-dimensional Euclidean space by the equation<br />
\[<br />
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1<br />
\]<br />
where the axes satisfy the inequality \(a>b>c>0\). When a particle with mass \(m\) moves freely on this surface, it is subject to a force that is always perpendicular to the tangent plane, and whose direction is, therefore, given by the vector \(\left(x / a^{2}, y / b^{2}, z / c^{2}\right)\) times some factor \(\lambda\) to be determined. (i) Show that the equations of motion then become<br />
\(\frac{d u}{d t}=-\lambda \frac{x}{a^{2}}, \quad \frac{d v}{d t}=-\lambda \frac{y}{b^{2}}, \quad \frac{d w}{d t}=-\lambda \frac{z}{c^{2}}\)<br />
\(\frac{d x}{d t}=\frac{u}{m}, \quad \frac{d y}{d t}=\frac{v}{m}, \quad \frac{d z}{d t}=\frac{w}{m}\)<br />
where \((u, v, w)\) is the momentum. (ii) Show that by taking two derivatives with respect to time in \((1)\) and replacing the second derivatives of \((x, y, z)\) according to (2), one gets the condition,<br />
\[<br />
m \lambda=\frac{u^{2} / a^{2}+v^{2} / b^{2}+w^{2} / c^{2}}{x^{2} / a^{4}+y^{2} / b^{4}+z^{2} / c^{4}} .<br />
\]<br />
(iii) Show that if the initial position of the particle satisfies (1), and the initial momentum is tangential to (1), then the whole trajectory stays on the ellipsoid. (iv) Show that the quantity<br />
\[<br />
A=u^{2}+\frac{(x v-y u)^{2}}{a^{2}-b^{2}}+\frac{(x w-z u)^{2}}{a^{2}-c^{2}}<br />
\]<br />
and similar ones, \(B\) and \(C\), which are obtained by the cyclic permutation of the triples \((x, y, z),(u, v, w)\), and \((a, b, c)\) are first integrals. These three quantities are not independent since one has the relation \(A+B+C=\) \(u^{2}+v^{2}+w^{2}\), where the right-hand side is the kinetic energy which is in fact the Hamiltonian of the system. \(A, B\), and \(C\) are in involution.Mathematicshttps://mathsgee.com/41448/dimensional-ellipsoid-imbedded-dimensional-euclidean-equationSat, 02 Jul 2022 02:10:11 +0000Consider the two-dimensional autonomous system in the \((x, y)\) plane described by \[ \frac{d x}{d t}=y^{3}\left(x^{2}-1\right)(2+x y), \quad \frac{d y}{d t}=x^{3}\left(y^{2}-1\right)(2-x y) \]
https://mathsgee.com/41403/consider-dimensional-autonomous-system-plane-described-right
Consider the two-dimensional autonomous system in the \((x, y)\) plane described by<br />
\[<br />
\frac{d x}{d t}=y^{3}\left(x^{2}-1\right)(2+x y), \quad \frac{d y}{d t}=x^{3}\left(y^{2}-1\right)(2-x y)<br />
\]<br />
(i) Show that the fixed points are given by \(A(-1,-1), B(1,1), C(1,-1)\), \(D(-1,1), E(-2,1), F(1,2), G(2,-1), H(-1,-2)\) and \(I(0,0)\) in the \(x-y)\) plane.<br />
(ii) Show that the fixed points \(A, B, C, D\) are saddles with eigenvalues \(-6\) and \(+2\). The fixed points \(E, F, G, H\) are stable attractors whilst \(I\) is neutral.<br />
(iii) The phase space is clearly not compact. Show but it can be made so by a coordinate change to a four dimensional system \((u, v, w, z)\) where<br />
\[<br />
z=x^{-1}, u=y x^{-1}, w e=y^{-1}, v=x y^{-1} .<br />
\]<br />
A new time coordinate \(\tau\) can be introduced to simplify the system if we define it by \(d \tau / d t=c^{6}\). The system can be restored to two dimensions by examining its behaviour on the slice where \(z=0\) and \(w=0\). On this plane we have<br />
\[<br />
\begin{gathered}<br />
\frac{d z}{d \tau}=z u^{3}\left(l-z^{2}\right)\left(u+2 z^{2}\right) \\<br />
\frac{d u}{d \tau}=\left(u^{2}-z^{2}\right)\left(-u+2 z^{2}\right)+u^{4}\left(l-z^{2}\right)\left(u+2 z^{2}\right)<br />
\end{gathered}<br />
\]<br />
and the new system has the fixed points \(A-I\) within a bounded circular region. These fixed points are augmented by \(J-Q\) on the circular boundary. The separatrix diagram indicates the generic fate of any trajectory. For example, a trajectory lying in the cell CADB will wind around in a spiral clockwise, indicative of quasi-periodic, oscillatory behaviour. The motion of a generic trajectory through the cell complex can be determined and a discrete mapping set up to desribe the sequence of separatrix changes.Mathematicshttps://mathsgee.com/41403/consider-dimensional-autonomous-system-plane-described-rightSat, 02 Jul 2022 01:18:03 +0000What are unit vectors?
https://mathsgee.com/38146/what-are-unit-vectors
What are unit vectors?Mathematicshttps://mathsgee.com/38146/what-are-unit-vectorsSun, 20 Feb 2022 02:43:40 +0000Sketch a a polar coordinate plot of \[ r=1+2 \sin 3 \theta, 0 \leq \theta \leq 2 \pi . \]
https://mathsgee.com/37383/sketch-a-polar-coordinate-plot-of-sin-3-theta-leq-theta-leq-pi
(a) Sketch a a polar coordinate plot of<br />
\[<br />
r=1+2 \sin 3 \theta, 0 \leq \theta \leq 2 \pi .<br />
\]<br />
<br />
(b) How many points lie in the intersection of the two polar graphs<br />
\[<br />
r=1+2 \sin 3 \theta, 0 \leq \theta \leq 2 \pi<br />
\]<br />
and<br />
\[<br />
r=1 \text { ? }<br />
\]<br />
(c) Algebraically find all values of \(\theta\) that<br />
\[<br />
1=1+2 \sin 3 \theta, 0 \leq \theta \leq 2 \pi .<br />
\]<br />
(d) Explain in a sentence or two why the answer to part (b) differs from (or is the same as) the number of solutions you found in part (c).Mathematicshttps://mathsgee.com/37383/sketch-a-polar-coordinate-plot-of-sin-3-theta-leq-theta-leq-piFri, 04 Feb 2022 11:58:45 +0000Sketch a polar coordinate plot of: \(r=2 \cos 3 \theta\)
https://mathsgee.com/37381/sketch-a-polar-coordinate-plot-of-r-2-cos-3-theta
Sketch a polar coordinate plot of:<br />
<br />
(a) \(r=2 \cos 3 \theta\)<br />
(b) \(r^{2}=-4 \sin 2 \theta\)<br />
(c) \(r=2 \sin \theta\)<br />
(d) \(r=2 \cos \theta\)<br />
(e) \(r=4+7 \cos \theta\)Mathematicshttps://mathsgee.com/37381/sketch-a-polar-coordinate-plot-of-r-2-cos-3-thetaFri, 04 Feb 2022 11:56:39 +0000What are standard unit vectors?
https://mathsgee.com/36647/what-are-standard-unit-vectors
What are standard unit vectors?Mathematicshttps://mathsgee.com/36647/what-are-standard-unit-vectorsTue, 25 Jan 2022 23:55:52 +0000How can we better coordinate national and international research efforts in agriculture?
https://mathsgee.com/35010/coordinate-national-international-research-agriculture
How can we better coordinate national and international research efforts in agriculture?Agriculturehttps://mathsgee.com/35010/coordinate-national-international-research-agricultureSun, 09 Jan 2022 02:01:22 +0000In coordinate geometry ___ are used to find the measurements of geometric figures.
https://mathsgee.com/24717/coordinate-geometry-used-measurements-geometric-figures
In coordinate geometry ___ are used to find the measurements of geometric figures.Mathematicshttps://mathsgee.com/24717/coordinate-geometry-used-measurements-geometric-figuresSun, 07 Feb 2021 03:06:36 +0000In which quadrant does the point (3, -4) lie in the coordinate plane?
https://mathsgee.com/24714/in-which-quadrant-does-the-point-lie-in-the-coordinate-plane
In which quadrant does the point (3, -4) lie in the coordinate plane?Mathematicshttps://mathsgee.com/24714/in-which-quadrant-does-the-point-lie-in-the-coordinate-planeSun, 07 Feb 2021 03:01:41 +0000Given the curve $9x^2+4y^2=36$, identify and then sketch the curve indicating the important points
https://mathsgee.com/10427/given-curve-identify-sketch-curve-indicating-important-points
Given the curve $9x^2+4y^2=36$, identify and then sketch the curve indicating the important pointsMathematicshttps://mathsgee.com/10427/given-curve-identify-sketch-curve-indicating-important-pointsMon, 11 May 2020 20:34:52 +0000Given $9y^2 + 4z^2 = x^2 + 36$...
https://mathsgee.com/8216/given-9y-2-4z-2-x-2-36
<p>Given $9y^2 + 4z^2 = x^2 + 36$</p>
<ol>
<li><strong>Rewrite the equation in standard form</strong></li>
<li><strong>Identify the surface</strong></li>
<li><strong>Draw each trace for the coordinate planes and describe the conic section. Label and scale your axes.</strong></li>
<li><strong>Graph ${9y^2 + 4z^2 = x^2 + 36}$. Label and scale the axes.</strong></li>
</ol>Mathematicshttps://mathsgee.com/8216/given-9y-2-4z-2-x-2-36Sat, 25 Apr 2020 16:55:27 +0000