A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $\overrightarrow{O P}$ to a point $P^{\prime}$ of the cylinder. Is the image of the Arctic Circle congruent to the image of the equator?
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5 degrees north of the Equator. Show more…
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A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder. Are distances near the equator distorted more than or less than distances near the Arctic Circle?
Transformations
Mappings and Functions
A piece of paper is wrapped around a globe of the Earth to form a cylinder as shown. $O$ is the center of the Earth and a point $P$ of the globe is projected along $O \vec{P}$ to a point $P^{\prime}$ of the cylinder. Mapping $M$ maps points $A$ and $B$ to the same image point. Explain why the mapping $M$ does not preserve distance.
The latitude and longitude of a point $P$ in the Northern Hemisphere are related to spherical coordinates $\rho, \theta, \phi$ as follows. We take the origin to be the center of the earth and the positive $z$ -axis to pass through the North Pole. The positive $x$ -axis passes through the point where the prime meridian (the meridian through Greenwich, England) intersects the equator. Then the latitude of $P$ is $\alpha=90^{\circ}-\phi^{\circ}$ and the longitude is $\beta=360^{\circ}-\theta^{\circ} .$ Find the great-circle distance from Los Angeles (lat. $34.06^{\circ} \mathrm{N},$ long. $118.25^{\circ} \mathrm{W}$ ) to Montréal (lat. $45.50^{\circ} \mathrm{N},$ long. $73.60^{\circ} \mathrm{W} ) .$ Take the radius of the earth to be 3960 $\mathrm{mi}$ . (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.)
MULTIPLE INTEGRALS
Triple Integrals in Spherical Coordinates
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