Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Mathematics of radio engineering wikipedia, lookup

Vector space wikipedia, lookup

Euclidean vector wikipedia, lookup

Line (geometry) wikipedia, lookup

Bra–ket notation wikipedia, lookup

Basis (linear algebra) wikipedia, lookup

Karhunen–Loève theorem wikipedia, lookup

Cartesian tensor wikipedia, lookup

Transcript

Multivariable Calculus Summary 1-Vectors DEFINITION: An n-tuple or an n-dimensional vector, is a symbol of the form x1 , x2 , x3 , . . ., x . n DEFINITION: R is the set of all n-tuples of real numbers. 2 Example: R is the set of all ordered pairs of real numbers. It is represented by the Cartesian plane. 3 R is the set of all ordered triples of real numbers. It is called the 3-dimensional vector space. Physical concept of a vector: a vector is a directed arrow that begins at some initial point called the tail and ends at some terminal point called the head. 2 Vectors in standard position or position vectors in R . An ordered pair numbers represents a vector v a, b of real from the origin (0, 0) to the point (a, b). a and b are called the components of the vector. 3 Position Vectors in R . An ordered triple a, b, c of real numbers represents a vector from the origin (0, 0, 0) to the point (a, b, c). n n Vector in R . An n-tuple of real numbers represents a vector in R . Definition: The norm of a vector | v |= x12 x 22 x32 .... x 2n 2 Note: In R If If v = x 1, x 2 , x 3 , . . ., xn is defined by and R3 , the norm of a vector is given by the length of the vector. v = a, b then its length of norm is given by | v |= a 2 b 2 v = a, b, c then its length of norm is given by | v |= a 2 b 2 c 2 Definition: A unit vector is a vector of length 1 or norm 1. To find the unit vector in the 1 v , multiply the vector by the scalar v 1 vector, then u = v is a unit vector in the direction of v v direction of a give vector -1- . Therefore, if v is any Operations of vector in Addition: a1 , b1 , c1 a2 , b2 , c2 a1 a2 , b1 b2 , c1 c2 Addition of vectors follows the parallelogram law of addition a1 , b1 , c1 a2 , b2 , c2 a1 a2 , b1 b2 , c1 c2 Scalar Multiplication: if k is a scalar and v a, b, c is a vector, then the scalar multiplication k v is given by the vector k v ka, kb, kc Note: k v is a vector parallel to v in the same direction if k>0, in the opposite direction if k<0. The length of k v is equal to the length of v multiply by |k| Note: v1 v2 v1 (1v2 ) Note: Vectors u and v are parallel if an only if u kv where k is a scalar. 2 Unit vectors in R : i 1, 0 and j 0, 1 Subtraction: Unit vectors in R 3 : i 1, 0, 0 and j 0, 1, 0 and k 0, 0, 1 2 Direction of a vector in R : The direction of a vector v a, b can be indicated by the b , and is the angle in standard position. a Theorem: If (same as ) and are the angles that the vector v a, b x-axis and y-axis respectively, then the unit vector in the direction of v is given by the vector u cos , cos i cos j cos angle where tan makes with R 3 : The direction of a vector v a, b, c can be indicated by the , , that the vector v a, b makes with x-axis, y-axis and z-axis Direction of a vector in angles respectively. v a, b, c makes with x-axis, y-axis and z-axis respectively, then the unit vector in the direction of v is given by the vector u cos , cos , cos i cos j cos k cos Theorem: If , , are the angles that the vector 2 Ways of expressing a vector in R : v a, b 2. Algebraic form: v ai bj 3. Polar form: v v , 4. Trigonometric form: v v i cos j sin or v v i cos j cos 1. Vector form: -2- Ways of expressing a vector in R3 : v a, b, c 2. Algebraic form: v ai bj ck 3. Trigonometric form: v v i cos j cos k cos Vector from points P1 to P2 1. Vector form: The vector from P1 ( x1 , y1 ) to P2 ( x2 , y2 ) is given by x2 x1 , y2 y1 The vector from P1 ( x1 , y1 , z1 ) to P2 ( x2 , y2 , z2 ) is given by x2 x1 , y2 y1 , z2 z1 Distance formulas: The distance between two points in In R 3 , distance is given by D x x y x x y y z z R2 is given by D 2 2 2 2 1 1 2 2 2 1 2 2 1 Midpoint formulas: The midpoint formula in The midpoint formula in x1 x2 y1 y2 , 2 2 x x y y2 z1 z2 , R 3 is given by M= 1 2 , 1 2 2 2 R2 is given by M= Circles and spheres: 2 In R the equation of a circle with center (h, k) and radius r is given by x h 2 ( y k )2 r 2 R 3 the equation of a sphere with center (h, k, L) and radius r is given by x h2 ( y k )2 ( z L)2 r 2 2 2 2 3 Note: In R , x h ( y k ) r represents a right circular cylinder In Dot product, scalar product or inner product: Definition: a1 , b1 a2 , b2 a1a2 b1b2 a1 , b1 , c1 a2 , b2 , c2 a1a2 b1b2 c1c2 Note: the dot product of two vectors is not a vector, it is a scalar. Theorem: If and u and v are vectors in then u v u v cos , where is the angle betweenu and v -3- y1 2 Theorem: Two vectors are perpendicular if and only if their dot product is 0. n Note: the above definitions can be generalize to R Properties of the dot product. If u, v and w are vectors and k is a scalar then 1. u u u , that is, u u u 2. u u 0 3. u v 0 if and only if 90 , that is, u and v are perpendicular (orthogonal) vectors 4. u v 0 if and only if 90, that is is acute 5. u v 0 if and only if 90 180, that is is obtuse 6. u v v u , dot product is commutative 7. u v w u v u w , dot product is distributive over addition 8. k u v ku v u kv , associativity of scalar and dot product 2 Projection: The orthogonal projection of a vector A on a vector B, is given by the component of A parallel to B. B A C D Let u AB, let v AD, then the projection of u on v is given by Pr ojv u Comv uAC Theorem: If u and v are vectors, then the projection of u on v is given by the vector u v u v 2 v. The length of the projection is given by Pr ojv u v v The vector w= u - Pr ojv u is called the component of u orthogonal to v. Pr ojv u -4-