SREENIVASA INSTITUTE of TECHNOLOGY and MANAGEMENT STUDIES (autonomous) (ENGINEERING MATHEMATICS-III) Course Material II- B.TECH / I - SEMESTER regulation: r18 Course Code: 18SAH211 Compiled by Department OF MATHEMATICS Unit-I: Numerical Integration Source:https://www.intmath.com/integration/integration-intro.php Numerical Integration: Simpson’s 1/3- Rule Note: While applying the Simpson’s 1/3 rule, the number of sub-intervals (n) should be taken as multiple of 2. Simpson’s 3/8- Rule Note: While applying the Simpson’s 3/8 rule, the number of sub-intervals (n) should be taken as multiple of 3. Numerical solution of ordinary differential equations Taylor’s Series Method Picard’s Method Euler’s Method Runge-Kutta Formula UNIT-II Multiple Integrals 1. Double Integration Evaluation of Double Integration Triple Integration UNIT-III Partial Differential Equations Partial differential equations are those equations which contain partial differential coefficients, independent variables and dependent variables. The independent variables are denoted by x and y and dependent variable by z. the partial differential coefficients are denoted as follows The order of the partial differential equation is the same as that of the order of the highest differential coefficient in it. UNIT-IV Vector Differentiation Elementary Vector Analysis Definition (Scalar and vector): Scalar is a quantity that has magnitude but not direction. For instance mass, volume, distance Vector is a directed quantity, one with both magnitude and direction. For instance acceleration, velocity, force Basic Vector System Magnitude of vectors: Let P = (x, y, z). Vector OP P is defined by OP p x i + y j + z k []x, y, z with magnitude (length) OP p x2 + y 2 + z2 Calculation of Vectors Vector Equation Two vectors are equal if and only if the corresponding components are equals Let a a1i + a2 j + a3 k and b b1i + b2 j + b3 k. Then a b a b , a b , a b 1 1 2 2 3 3 Addition and Subtraction of Vectors a b (a1 b1)i +(a2 b2 ) j +(a3 b3)k Multiplication of Vectors by Scalars If is a scalar, then b (b )i + (b ) j + (b )k 1 2 3 Example 1 Given p 5i + j - 3k and q 4i - 3j + 2k . Find a) p + q b) p - q cp) Magnitude of vector d) 2 q - 10 p Vector Products If aaiajak1 + 2 + 3 and bbibjbk 1 + 2 + 3 , ~ ~ ~ ~ ~ ~ ~~ 1) Scalar Product (Dot product) a b a1 b 1 + a 2 b 2 + a 3 b 3 ~~ or a.b | a || b | cos, is the angle between a and b ~ ~ ~ ~ 2) Vector Product (Cross product) i j k ~~~ a b a1 a 2 a 3 ~~ b1 b 2 b 3 ab23 - abi 32 - ab 13 - abj 31 + ab 12 - abk 21 ~~ ~ Differentiation of Two Vectors Del Operator Or Nabla (Symbol ) Operator is called vector differential operator, defined as i + j + k . x ~ y ~ z ~ Grad (Gradient of Scalar Functions) If x,y,z is a scalar function of three variables and f is differentiable, the gradient of f is defined as grad i + j + k . ~ ~ x y ~ z * is a scalar function * is a vector function Example.2 If x2 yz3 + xy2 z 2 , determine grad at P (1,3,2). Solution Given x2 yz3 + xy2 z 2 , hence 2xyz3 + y 2 z 2 x x2 z3 + 2xyz2 y 3x2 yz2 + 2xy2 z z Therefore, i + j + k x ~ y ~ z ~ (2xyz3 + y 2 z 2 ) i + (x2 z3 + 2xyz2 ) j ~ ~ + (3x2 yz2 + 2xy2 z) k . ~ At P (1,3,2), we have (2(1)(3)(2)3 + (3)2 (2)2 ) i + ((1)2 (2)3 + 2(1)(3)(2)2 ) j ~ ~ + (3(1)2 (3)(2)2 + 2(1)(3)2 (2))k . ~ 84i + 32 j + 72k . ~ ~ ~ Example.2 3 2 3 If x yz+ xy z , determine grad at point P (1,2,3). Given x3 yz + xy2 z3 , then x y z Grad At P (1,2,3), 126i +111 j +110k . ~ ~ ~ Grad Properties If A and B are two scalars, then 1) (A+ B) A+ B 2) (AB) A(B) + B(A) Divergence of a Vector If A ax i + ay j + az k, the divergence of A is ~ ~ ~ ~ ~ defined as div A . A ~ ~ i + j + k .(ax i + ay j + az k) x ~ y ~ z ~ ~ ~ ~ a a a div A . A x + y + z . ~ ~ x y z Example.1 2 2 If A x y i - xyz j + yz k, ~ ~ ~ ~ determine div A at point (1,2,3). ~ Answer a a a div A . A x + y + z ~ ~ x y z 2xy- xz + 2yz. At point (1,2,3), div A 2(1)(2) - (1)(3) + 2(2)(3) ~ 13. Example.2 3 2 2 3 If A x y i + xy z j - yz k, ~ ~ ~ ~ determine div A at point (3,2,1). ~ a a a Answer div A . A x + y + z ~ ~ x y z At point (3,2,1), div A ~ 114. Remarks A is a vector function, but div A is a scalar function. ~ ~ If div A 0, vector A is called solenoidvector. ~ ~ Curl of a Vector If A ax i + ay j + az k, the curl of A is defined by ~ ~ ~ ~ ~ curl A A ~ ~ i + j + k (ax i + ay j + az k) x ~ y ~ z ~ ~ ~ ~ i j k ~ ~ ~ curl A A . ~ ~ x y z ax ay az Example 1 4 2 2 2 2 2 If A (y - x z ) i + (x + y ) j - x yzk, ~ ~ ~ ~ determine curl A at (1,3,-2). ~ Solution i j k ~ ~ ~ curl A A ~ ~ x y z y 4 - x2 z 2 x2 + y 2 - x2 yz 2 2 2 (-x yz) - (x + y ) i y z ~ - (-x2 yz) - (y4 - x2 z 2 ) j x z ~ 2 2 4 2 2 + (x + y ) - (y - x z )k x y ~ -x2 z i - (-2xyz+ 2x2 z) j + (2x - 4y3 ) k . ~ ~ ~.