MAT 21C Lecture Notes - Lecture 21: Differentiable Function, Unit Vector, Dot Product
MAT 21C – Lecture 21 – Differentiability of Multi-Variable Functions
• Summary of Topics
1. Power Series (Chapter 10)
a. A power series takes on the form
which has
radius of convergence, and center, a.
b. The series converges (absolutely) if and diverges if
. Use the ratio or root test to find R and then check
the endpoints separately.
c. Taylor Series: f(x) has Taylor series at x = a represented by
. The Taylor Polynomial of f(x) at x = a of
order, n is
.
d. Taylor’s Reaider Theore states that
where
for some c between a
and x. If for all such c, then
.
e. The Taylor Series of f(x) at x = a converges to f(x) inside
if as .
2. Vectors (Chapter 12)
a. If
and
are two vectors,
then it has a right-handed unit vector,
which is orthogonal to
vectors
and .
b. Dot Product:
c. Cross Product:
.
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MAT 21C Full Course Notes
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Document Summary
=(cid:2868) radius of convergence, 0 and center, a: the series converges (absolutely) if |(cid:1876) |< and diverges if. Use the ratio or root test to find r and then check the endpoints (cid:1876)= separately. (cid:4666)(cid:1876) (cid:4667). The taylor polynomial of f(x) at x = a of (cid:4666)(cid:4667)(cid:4666)(cid:4667)! =(cid:2868: taylor"s re(cid:373)ai(cid:374)der theore(cid:373) states that (cid:4666)(cid:1876)(cid:4667)= (cid:1842)(cid:4666)(cid:1876)(cid:4667)+(cid:4666)(cid:1876)(cid:4667) where (cid:4666)(cid:1876)(cid:4667)= (cid:4666)(cid:1876) (cid:4667)+(cid:2869) for some c between a (cid:4666)+(cid:3117)(cid:4667)(cid:4666)(cid:4667) Scalar triple product: (cid:4666)(cid:1873) (cid:1876) (cid:1874) (cid:4667) (cid:1875) = volume of the parallelepiped = |(cid:1873)(cid:2869) (cid:1873)(cid:2870) (cid:1873)(cid:2871) (cid:1874)(cid:2869) (cid:1874)(cid:2870) (cid:1874)(cid:2871) (cid:1875)(cid:2869) (cid:1875)(cid:2870) (cid:1875)(cid:2871)|. point (cid:1842)(cid:2868)(cid:4666)(cid:1876)(cid:2868),(cid:1877)(cid:2868),(cid:1878)(cid:2868)(cid:4667) in a direction (cid:1874) is (cid:4666)(cid:1872)(cid:4667)= (cid:2868) +(cid:1872)(cid:1874) where = Then, {(cid:1876)= (cid:1876)(cid:2868)+(cid:1872)(cid:1874)(cid:2869) (cid:1841)(cid:1842) and (cid:2868) =(cid:1841)(cid:1842)(cid:2868) (cid:1877)= (cid:1877)(cid:2868)+(cid:1872)(cid:1874)(cid:2870) (cid:1878)= (cid:1878)(cid:2868)+(cid:1872)(cid:1874)(cid:2871: the cartesian for the plane through point (cid:1842)(cid:2868)(cid:4666)(cid:1876)(cid:2868),(cid:1877)(cid:2868),(cid:1878)(cid:2868)(cid:4667) with normal vector, (cid:1866) is (cid:1866) (cid:4666) (cid:2868) (cid:4667)=0. If (cid:1866) =(cid:1827)(cid:2835) +(cid:1828)(cid:2836) +(cid:1829) , then ax. = (cid:4666)(cid:1876)(cid:2868)(cid:4667)+(cid:4666) (cid:4667) : suppose f(x) is differentiable at (cid:1876)(cid:2868).
