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Chapter 5

MAT 211 Chapter 5: matt211 textbook ch5

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Mathematics
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MAT 211
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Course Summary Math 211 table of contents I. Functions of several variables. n II. R . III. Derivatives. IV. Taylor’s Theorem. V. Di▯erential Geometry. VI. Applications. 1. Best a▯ne approximations. 2. Optimization. 3. Lagrange multipliers. 4. Conservation of energy. I. Functions of several variables. De▯nition 1.1. Let S and T be sets. The Cartesian product of S and T is the set of ordered pairs: S ▯ T := f(s;t) j s 2 S;t 2 Tg: De▯nition 1.2. Let S and T be sets. A function from S to T is a subset W of the Cartesian product S ▯T such that: (i) for each s 2 S there is an element in W whose ▯rst component is s, i.e., there is an element (s;t) 2 W for some t 2 T; and (ii) if (s;t) and (s;t ) are in W, then t = t . Notation: if (s;t) 2 W, we write f(s) = t. The subset W, which is by de▯nition the function f, is also called the graph of f. De▯nition 1.3. Let f:S ! T be a function between sets S and T. 1. f is one-to-one or injective if f(x) = f(y) only if x = y. 2. The image or range of f is ff(s) 2 T j s 2 Sg. The image will be denoted by im(f) or f(S). 3. f is onto if im(f) = T. 4. The domain of f is S and the codomain of f is T. 5. The inverse image of t 2 T is f ▯1 (t) := fs 2 S j f(s) = tg. 1 De▯nition 1.4. Let f:S ! T and g:T ! U. The composition of f and g is the function g ▯ f:S ! U given by (g ▯ f)(s) := g(f(s)). De▯nition 1.5. R is the Cartesian product of R with itself n times. We think of R as n the set of ordered n-tuples of real numbers: R := f(a ;1::;a )nj a 2iR;1 ▯ i ▯ ng: The elements of R are called points or vectors. De▯nition 1.6. A function of several variables is a function of the form f:S ! R m where n S ▯ R . Writing f(x) = (f (x1;:::;f (x)m, the function f :S i R, for each i = 1;:::;m, is called the i-th component function of f. De▯nition 1.7. Let f be a function of several variables, f:S ! R , with S ▯ R . If n = 1, then f is a parametrized curve, if n = 2, then f is a parametrized surface. In general, we say f is a parametrized n-surface. m m De▯nition 1.8. A vector ▯eld is a function of the form f:S ! R where S ▯ R . n De▯nition 1.9. If f:S ! R with S ▯ R , a level set of f is the inverse image of a point in R. A drawing showing several level sets is called a contour diagram for f. II. R . linear structure. n De▯nition 2.1. The i-th coordinate of a = (a ;::1;a ) 2 n is a . Forii = 1;:::;n, de▯ne the i-th standard basis vector for R to be the vector e whose coordinates are all zero except i the i-th coordinate, which is 1. De▯nition 2.2. The additive inverse of a = (a ;:::1a ) 2 n n is the vector ▯a := (▯a 1:::;▯a )n n De▯nition 2.3. In R , de▯ne 0 := (0;:::;0), the vector whose coordinates are all 0. n De▯nition 2.4. (Linear structure on R .) If a = (a ;::1;a ) and b = (b ;:::1b ) arn points in R and s 2 R, de▯ne a + b = (a 1:::;a n + (b 1:::;b n := (a + b1;:::1a + b )n n sa = s(a 1:::;a n := (sa ;::1;sa ): n The point a + b is the translation of a by b (or of b by a) and sa is the dilation of a by a factor of s. De▯ne a ▯ b := a + (▯b). 2 metric structure. n n n De▯nition 2.5. The dot product on R is the function R ▯ R ! R given by Xn (1 ;:::;n ) ▯1(b ;:n:;b ) :ai i: i=1 n The dot product is also called the inner product or scalar product. If a;b 2 R , the dot product is denoted by a ▯ b, as above, or sometimes by (a;b) or ha;bi. De▯nition 2.6. The norm or length of a vector a =1(a ;:n:;a ) 2 R is v u n p t X 2 jaj := a ▯ a = ai: i=1 The norm can also be denoted by jjajj. n De▯nition 2.7. The vector a 2 R is a unit vector if jaj = 1. n De▯nition 2.8. Let p 2 R and r 2 R. 1. The open ball of radius r centered at p is the set n Br(p) := fa 2 R j ja ▯ pj < rg: 2. The closed ball of radius r centered at p is the set Br(p) := fa 2 R j ja ▯ pj ▯ rg: 3. The sphere of radius r centered at p is the set n Sr(p) := fa 2 R j ja ▯ pj = rg: n De▯nition 2.9. The distance between a = 1a ;::n;a ) and b 1 (b ;n::;b ) in R is v u Xn t 2 d(a;b) := ja ▯ bj = (ai▯ bi) : i=1 De▯nition 2.10. Points a;b 2 R are perpendicular or orthogonal if a ▯ b = 0. De▯nition 2.11. Suppose a;b are nonzero vectors in R . The angle between them is de▯ned a▯b to be cos1jajjbj 3 De▯nition 2.12. Let a;b 2 R with b 6= 0. The component of a along b is the scalar a▯b a▯b c := b▯b= jbj. The projection of a along b is the vector cb where c is the component of a along b. affine subspaces. De▯nition 2.13. A nonempty subset W ▯ R is a linear subspace if it is closed under vector addition and scalar multiplication. This means that: (i) if a;b 2 W then a + b 2 W, and (ii) if a 2 W and s 2 R, then sa 2 W. De▯nition 2.14. A vector v 2 R is a linear combination of vectors v ;:1:;v 2 k if there P k are scalars 1 ;:::;ak2 R such that v = i=1ai i. De▯nition 2.15. A subspace W ▯ R is spanned by a subset S ▯ R if every element of W can be written as a linear combination of elements of S. If W is spanned by S, we write span(S) = W. De▯nition 2.16. The dimension of a linear subspace W ▯ R is the smallest number of vectors needed to span W. De▯nition 2.17. Let W be a subset of R and let p 2 R . The set p + W := fp + w j w 2 Wg n is called the translation of W by p. An a▯ne subspace of R is any subset of the form p+W where W is a linear subspace of R . In this case, the dimension of the a▯ne subspace is de▯ned to be the dimension of W. n De▯nition 2.18 A k-plane in R is an a▯ne subspace of dimension k. A line is a 1-plane, and a hyperplane is a (n ▯ 1)-plane. affine functions. De▯nition 2.19. A function L:R ! Rn m is a linear function (or transformation or map) if n it preserves vector addition and scalar multiplication. This means that for all a;b 2 R and for all s 2 R, 1. L(a + b) = L(a) + L(b); 2. L(sa) = sL(a). De▯nition 2.20. (Linear structure on the space of linear functions.) Let L and M be linear functions with domain R and codomain R . m n m 1. De▯ne the linear function L + M:R ! R by (L + M)(v) := L(v) + M(v) for all v 2 R . 4 2. If s 2 R, de▯ne the linear function sL:R ! R m by (sL)(v) := L(sv) n for all v 2 R . n m De▯nition 2.21. A function f:R ! R is an a▯ne function (or transformation or map) if it is the ‘translation’ of a linear function. This means that there is a linear funtion L:R ! R m and a point p 2 R m such that f(v) = p + L(v) for all v 2 R . n De▯nition 2.22. Let W be a k-dimensional a▯ne subspace of R . A parametric equation for W is any a▯ne function f:R ! R whose image is W. De▯nition 2.23. An m▯n matrix is a rectangular block of real numbers with m rows and n columns. The real number appearing in the i-th row and j-th column is called the i;j-th entry of the matrix. We write A = (a ) for the matrix whose i;j-th entry is a . ij ij De▯nition 2.24. (Linear structure on matrices.) Let A = (a ) anijB = (b ) be mij n matrices. De▯ne A + B := (a +ij ).ijf s 2 R, de▯ne sA := (sa ).ij De▯nition 2.25. (Multiplication of matrices.) Let A = (a ) ij an m ▯ k matrix, and let B = (b ijbe a k ▯ n matrix. De▯ne the product, AB to be the m ▯ n matrix whose i;j-th P k entry is ‘=1ai‘ ‘j De▯nition 2.26. Let A = (a ) ij an m▯n matrix. The linear function determined by (or associated with) A is the function L :R ! R m such that A P n P n L(x 1:::;x n = ( a1j j:::; amjxj): j=1 j=1 De▯nition 2.27. Let L:R n ! R m be a linear function. The matrix determined by (or associated with) L is the m ▯ n matrix whose i-th column is the image of the i-th standard n basis vector for R under L, i.e., L(i ). De▯nition 2.28. An n▯n matrix, A, is invertible or nonsingular if there is an n▯n matrix B such that AB = I whnre I is nhe identity matrix whose entries consist of 1s along the ▯1 diagonal and 0s otherwise. In this case, B is called the inverse of A and denoted A . theorems n Theorem 2.1. Let a;b;c 2 R and s;t 2 R. Then 1. a + b = b + a. 2. (a + b) + c = a + (b + c). 3. 0 + a = a + 0 = a. 5 4. a + (▯a) = (▯a) + a = 0. ~ 5. 1a = a and (▯1)a = ▯a. 6. (st)a = s(ta). 7. (s + t)a = sa + ta. 8. s(a + b) = sa + sb. Theorem 2.2. Let a;b;c 2 R and s 2 R. Then 1. a ▯ b = b ▯ a. 2. a ▯ (b + c) = a ▯ b + a ▯ c. 3. (sa) ▯ b = s(a ▯ b). 4. a ▯ a ▯ 0. 5. a ▯ a = 0 if and only if a = 0. n Theorem 2.3. Let a;b 2 R and s 2 R. Then 1. jaj ▯ 0. 2. jaj = 0 if and only if a = 0. 3. jsaj = jsjjaj. 4. ja ▯ bj ▯ jajjbj (Cauchy-Schwartz inequality). 5. ja + bj ▯ jaj + jbj (triangle inequality). Theorem 2.4. Let a;b 2 R be nonzero vectors. Then a ▯ b ▯1 ▯ ▯ 1: jajjbj This shows that our de▯nition of angle makes sense. n Theorem 2.5. (Pythagorean theorem.) Let a;b 2 R . If a and b are perpendicular, then jaj + jbj = ja + bj . n Theorem 2.6. Any linear subspace of R is spanned by a ▯nite subset. 6 Theorem 2.7. If a = (a ;1::;a )n6= 0 and p = (p 1:::;p n are elements of R , then H := fx 2 R j (x ▯ p) ▯ a = 0g is a hyperplane. In other words, the set of solutions, (1 ;:::;n ), to the equation1a1x + P n ▯▯▯ + n xn= d where d = i=1ai iis a hyperplane. Conversely, every hyperplane is the set of solutions to an equation of this form. Theorem 2.8. If L:R ! R m is a linear function and W ▯ R is a linear subspace, then m L(W) is a linear subspace of R . Theorem 2.9. A linear map is determined by its action on the standard basis vectors. In other words: if you know the images of the standard basis vectors, you know the image of an arbitrary vector. Theorem 2.10. The image of the linear map determined by a matrix is the span of the columns of that matrix. Theorem 2.11. Let W be a k-dimensional subspace of R spanned by vectors v ;:::;v1, k n and let p 2 R . Then a parametric equation for the a▯ne space p + W is k n f:R ! R Xk (a1;:::;ak) 7! p + ai i: i=1 Theorem 2.12. Let L be a linear function and let A be the matrix determined by L. Then the linear map determined by A is L. (The converse also holds, switching the roles of L and A.) Theorem 2.13. The linear structures on linear maps and on their associated matrices are combatible: Let L and M be linear functions with associated matrices A and B, respectively, and let s 2 R. Then the matrix associated with L + M is A + B, and the matrix associated with sL is sA. n k k m Theorem 2.14. Let L:R ! R and M:R ! R be linear functions with associated matrices A and B, respectively. Then the matrix associated with the composition, M ▯ L is the product BA. III. Derivatives. De▯nition 3.1. A subset U ▯ R is open if for each u 2 U there is a nonempty open ball centered at u contained entirely in U: there exists a real number r > 0 such thrt B (u) ▯ U. 7 n n De▯nition 3.2. A point u 2 R is a limit point of a subset S ▯ R if every open ball centered at u, B (u), contains a points of S di▯erent from u. r De▯nition 3.3. Let f:S ! R m be a function with S ▯ R . Let s be a limit point of S. The m limit of f(x) as x approaches s is v 2 R if for all real numbers ▯ > 0, there is a real number ▯ > 0 such that 0 < jx ▯ sj < ▯ and x 2 S ) jf(x) ▯ vj < ▯. Notation: lim x!s f(x) = v. De▯nition 3.4. Let f:S ! R m with S ▯ R , and let s 2 S. The function f is continuous at s 2 S if for all real numbers ▯ > 0, there is a real number ▯ > 0 such that jx ▯ sj < ▯ and x 2 S ) jf(x) ▯ f(s)j < ▯. (Thus, f is continuous at a limit point s 2 S if and only if limx!s f(x) = f(s) and f is automatically continuous at all points in S which are not limit points of S.) The function f is continuous on S if it is continuous at each point of S. m n De▯nition 3.5. Let f:U ! R with U an open subset of R , and let eibe the i-th standard basis vector for R . The i-th partial of f at u 2 U is the vector in R @f f(u + tei) ▯ f(u) @x (u) := t!0 t i provided this limit exists. De▯nition 3.6. Let f:U ! R with U an open subset of R . Let u 2 U, and let v 2 R be n a unit vector. The directional derivative of f at u in the direction of v is the real number f(u + tv) ▯ f(u) fv(u) := lim t!0 t provided this limit exists. The directional derivative of f at u in the direction of an arbitrary nonzero vector w is de▯ned to be the directional derivative of f at u in the direction of the unit vector w=jwj. De▯nition 3.7. Let f:U ! R m with U an open subset of R . Then f is di▯erentiable at n m u 2 U if there is a linear function Dfu:R ! R such that jf(u + h) ▯ f(u) ▯ Df (u)j lim = 0: h!0 jhj The linear function Df u is then called the derivative of f at u. The notation f (u) is sometimes used instead of Df . uhe function f is di▯erentiable on U if it is di▯erentiable at each point of U. De▯nition 3.8. Let f:U ! R m with U an open subset of R . The Jacobian matrix of f at u 2 U is the m ▯ n matrix of partial derivatives of the component functions of f: 2 @f @f 3 ▯ ▯ @x1(u) ::: @x1 (u) @f i 6 . . n. 7 Jf(u) := (u) = 4 . .. . 5 : @x j @fm @fm @x1(u) ::: @xn (u) 8 1. The i-th column of the Jacobian matrix is the i-th partial derivative f at u and is called the i-th principal tangent vector to f at u. 2. If n = 1, then f is a parametrized curve and the Jacobian matrix consists of a single column. This column is the tangent vector to f at u or the velocity of f at u, and its length is the speed of f at u. We write 0 0 0 f (u) = (1 (u);:::;m (t)) for this tangent vector. 3. If m = 1, the Jacobian matrix consists of a single row. This row is called the gradient vector for f at u and denoted rf(u) or gradf(u): ▯ ▯ @f @f rf(u) := (u);:::; (u) : @x 1 @xn theorems Theorem 3.1. Let f:S ! R m and g:S ! R m where S is a subset of R . 1. The limit of a function is unique. 2. The limit, lix!s f(x), exists if and only if the corresponding limits for each of the component functions, limx!s fi(x), exists. In that case, ▯ ▯ x!sm f(x) = x!s 1 (x);:::x!simmf (x) : 3. De▯ne f+g:S ! R by (f+g)(x) := f(x)+g(x). If lim f(x) = a and lim g(x) = x!s m x!s b, then limx!s(f + g)(x) = a + b. Similarly, if t 2 R, de▯ne tf:U ! Rby (tf)(x) := t(f(x)). If lim f(x) = a, then lim (tf)(x) = ta. x!s x!s 4. If m = 1, de▯ne (fg)(x) := f(x)g(x) and (f=g)(x) := f(x)=g(x) (provided g(x) 6= 0). If lim f(x) = a and lim g(x) = b, then lim (fg)(x) = ab and, if b 6= 0, then x!s x!s x!s limx!s (f=g)(x) = a=b. 5. If m = 1 and g(x) ▯ f(x) for all x, then limx!sg(x) ▯ lim x!sf(x) provided these limits exist. m m n Theorem 3.2. Let f:S ! R and g:S ! R where S is a subset of R . 1. The function f is continuous if and only if the inverse image of every open subset of m n R under f is the intersection of an open subset of R with S. 9 2. The function f is continuous at s if and only if each of its component functions is continuous at s. 3. The composition of continuous functions is continuous. 4. The functions f + g and tf for t 2 R as above are continuous at s 2 S provided f and g are continuous at s. 5. If m = 1 and f and g are continuous at s 2 S, then fg and f=g are continuous at s (provided g(s) 6= 0 in the latter case). 6. A function whose coordinate functions are polynomials is continuous. n m Theorem 3.3. If f:R ! R is a linear transformation, then f is di▯erentiable at each p 2 R , and Df =pf. k m Theorem 3.4. (The chain rule.) Let f:U ! R and g:V ! R where U is an open subset of R and V is an open subset of R . Suppose that f(U) ▯ V so that we can form the m composition, g ▯f:U ! R . Suppose that f is di▯erentiable at p 2 U and g is di▯erentiable at f(p); then g ▯ f is di▯erentiable at p, and D(g ▯ f) p Dg f(p) Df p In terms of Jacobian matrices, we have, J(g ▯ f)(p) = Jg(f(p))Jf(p): Theorem 3.5. Let f:U ! R m where U is an open subset of R . Then f is di▯erentiable at p 2 U if and only if each component function f :i ! R is di▯erentiable, and in that case, Df (v) = (Df (v);:::;Df (v)) for all v 2 R . p 1p mp Theorem 3.6. Let f:U ! R where U is an o
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