By Groen O., Naess A.
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Additional info for Einstein's theory: A rigorous introduction for the mathematically untrained
And a vector representing, say, the wind velocity at the North Pole may be thought of as an arrow in this tangent plane. Thirdly, finite distances can be defined in curved spaces. 7 Space and metric 62 distance we are going to find. 5. Chapter 2 Differential calculus This chapter is written for those people who have the courage to approach the mathematics of general relativity without being familiar with differential calculus. The use of this fabulous creation by Newton and Leibniz is essential and omnipresent on our way to Einstein’s field equations.
12) we get the component form of the vector A A = Ax ex + Ay ey + Az ez . 15) Also from Fig. 15, and using the Pythagorean theorem as we did in the two-dimensional case in Eq. 17) Taking the square root of both sides of Eq. 17), we get the magnitude of A in terms of its components |A| = (Ax )2 + (Ay )2 + (Az )2 . 5 Three and more dimensions 49 If two vectors A and B in a three-dimensional space are added to give a third vector C, we obtain a projection onto the (x, y) plane just like Fig. 8. 19) z z z C =A +B .
6) are termed the component vectors of A. In order to define these we must first consider products of scalars and vectors. If the scalar is equal to a number k, the product kA is defined as a vector with magnitude k|A|, and with the same direction as A. With, for example k = 2, the magnitude of the vector is doubled. If the scalar is a negative number, −k, the product −kA is defined as a vector with magnitude k|A|, and with opposite direction to that of A. If k = 0, we write kA = 0 and call 0 the null vector.