MATH 251 Lecture 1

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Multivariable Calculus

James Vargo
math.tamu.edur/~vargo/courses/m251.html
BLOC 620c

(No TA)

No calculators on exams.

Homework

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• Written homework (due each Wednesday)

The Celestial Sphere

Vectors and Trigonometry

Spherical Coordinates

Define a point in space using

• $\rho$ (distance to origin; radius; nonnegative integer)
• $\phi$ (colatitude = $\pi /2$ − latitude; angle made with $z$-axis; 0 to $\pi$)
• $\theta$ (longitude; angle made from $x$; 0 to $2\pi$)

Converting to Cartesian Coordinates

• $x=\rho \sin {\phi }\cos {\theta }$
• $y=\rho \sin {\phi }\sin {\theta }$
• $z=\rho \cos {\phi }$

What are the latitude/colatitude of College Station?

lat. = 30.6°

colat = 90° − lat = 59.4°

Assuming Earth is Fixed, what direction does the celestial sphere rotate? =

(left-hand rule)

In a day, how long is Capella above the horizon

For a star (Capella) in celestial sphere,

• Declimation = latitude from horizon (46°, so colatitude is 90-46 = 44°)
• Right-Ascension = longitude (we don't care about this)

The star makes a complete revolution once every 24 hours. If we slice the sphere along Capella's celestial path, we get a circle. If we find the angle $\theta$ that represents half of the angle at which Capella is above the horizon (from its rising to when it's directly overhead)

$24\times \left({\frac {\theta }{\pi }}\right)$

For the observational vector (straight up; ${\vec {v}}$ and the vector from Earth to the declination on the horizon ${\vec {w}}$:

${\vec {v}}\cdot {\vec {w}}=0$

We find that ${\vec {v}}=\left(\rho =1,\,\theta =0,\,\phi =59.4^{\circ }\right)=\left\langle \sin {59.4^{\circ }},\,0,\,\cos {59.4^{\circ }}\right\rangle$ and ${\vec {w}}=\left(\rho =1,\,\theta ,\,\phi =44^{\circ }\right)=\left\langle \sin {44^{\circ }}\cos {\theta },\,\sin {44^{\circ }}\sin {\theta },\,\cos {44^{\circ }}\right\rangle$

For a different problem (e.g. the sun), just change the declination ($\phi$ in ${\vec {w}}$)