Mathematics for Physical Sciences III Change of lecturer: First 4 weeks: myself again! Remaining 8 weeks: Dr Stephen O Sullivan Continuous Assessment Test Date to be announced (probably Week 7 or 8) - 30% of total mark Online Notes: http://www.maths.dit.ie/~dmackey/lectures.html Dana Mackey (DIT) Maths 1 / 18
First topics: 1 Vector fields and vector calculus 2 Differential operators: gradient, divergence and curl 3 Line integrals, surface integrals and Integral Theorems Dana Mackey (DIT) Maths 2 / 18
Motivation: Maxwell s equations Differential form D = ρ f B = 0 E = B t H = J f + D t S Integral form D da = Q f (V ) V B da = 0 V Edl = Φ S(B) S t H dl = I f,s + Φ S(D) t Dana Mackey (DIT) Maths 3 / 18
Vector calculus Vector calculus is concerned with differentiation and integration of vector fields, that is functions with many components, such as F(x,y) = (f (x,y),g(x,y)) or F(t) = (f (t),g(t),h(t)). Review of vectors A vector in the plane is a directed line segment (a quantity characterized by both magnitude and direction). Examples of vectors in physics include velocity, force, acceleration, etc. For a general vector v = (x,y) = xi + yj, the magnitude or length of v can be defined as v = x 2 + y 2 Dana Mackey (DIT) Maths 4 / 18
Vector functions of one variable A vector function or vector field of one variable is a rule that assigns a vector to each value of the variable. For example, when a particle moves through the plane during a time interval, each of the particle coordinates is a function of time: x = x(t) and y = y(t). The path of the particle is then described by the position vector r(t) = (x(t),y(t)) = x(t)i + y(t)j Note: To differentiate a vector function of one variable we simply differentiate each component with respect to that variable. For example, if r(t) = (x(t),y(t)) = x(t)i + y(t)j then dr dt = ( dx dt, dy dt ) = dx dt i + dy dt j. Dana Mackey (DIT) Maths 5 / 18
If r(t) is the position vector of a particle moving in the plane then we have v(t) = dr dt is the velocity of the particle; v(t) is the speed of the particle; v v is the direction of motion; a(t) = dv dt = d 2 r dt 2 is the acceleration of the particle. Dana Mackey (DIT) Maths 6 / 18
Example: In the case of a projectile launched from the origin (0,0), the path is described by the position vector r(t) = V 0 t cos(α)i + [V 0 t sin(α) 12 ] gt2 j (1) where V 0 is the initial (launch) velocity and α is the initial (launch) angle. Exercise: For the projectile motion defined by the position vector in Equation (1), calculate 1 The velocity, speed and direction of motion of the particle at each moment of time t; 2 The acceleration of the particle 3 The maximum height, flight time and range of the motion. Dana Mackey (DIT) Maths 7 / 18
Review of some vector operations Consider two vectors in 3-dimensional space: A = A 1 i + A 2 j + A 3 k and B = B 1 i + B 2 j + B 3 k. We define The dot product or inner product of A and B is A B = A 1 B 1 + A 2 B + A 3 B 3 = A B cos(θ) where θ is the angle between the vectors. The cross product of A and B is i j k A B = A B sin(θ) n = det A 1 A 2 A 3 B 1 B 2 B 3 where n is the unit vector in the direction which is perpendicular to the plane formed by A and B. Dana Mackey (DIT) Maths 8 / 18
Properties of the dot and cross products 1 The dot product of two vectors is a scalar (number) while the cross product is a vector; 2 If the dot product of two vectors is zero then the two vectors are perpendicular (the angle between them is 90 ); 3 If the cross product of two vectors is zero then the vectors are parallel (the angle between them is 0 ); 4 A B = B A but A B = B A. 5 i i = j j = k k = 0 6 i j = j i = k, j k = k j = i and k i = i k = j Example: Let A = 2i + j + k and B = 4i + 3j + k. Calculate A B and A B. Same problem for A = i 2j and B = 3i + 4j. Dana Mackey (DIT) Maths 9 / 18
Gradient, Divergence and Curl First, we define the vector operator (del) as = i x + j y + k z = ( x, y, z ) 1 The gradient of a scalar function Φ(x,y,z) is grad(φ) = Φ = i Φ x + j Φ y + k Φ z = ( Φ x, Φ y, Φ ) z 2 The divergence of a vector field A(x,y,z) = A 1 i + A 2 j + A 3 k is div(a) = A = A 1 x + A 2 y + A 3 z 3 The curl of A is i j k curl(a) = A = det x y z A 1 A 2 A 3 Dana Mackey (DIT) Maths 10 / 18
The curl of a vector can also be calculated with the formula ( A3 A = y A ) ( 2 A1 i + z z A ) 3 j + x ( A2 x A 1 y ) k Note: The gradient operator is applied to a scalar field and the result is a vector field. The divergence operator is applied to a vector field and the result is a scalar field. The curl operator is applied to a vector field and the result is also a vector field. Dana Mackey (DIT) Maths 11 / 18
Definition: The expression is called the Laplacian of Φ. ( Φ) = 2 Φ = 2 Φ x 2 + 2 Φ y 2 + 2 Φ z 2 Exercise: If Φ = x 2 yz 3 and A = xzi y 2 j + 2x 2 yk find (a) Φ; (b) 2 Φ; (c) A; (d) A; (e) div(φa); (f) curl(φa). Dana Mackey (DIT) Maths 12 / 18
Properties of the operator 1 (Φ 1 + Φ 2 ) = Φ 1 + Φ 2 (gradient of sum = sum of gradients) 2 (A + B) = A + B (divergence of sum = sum of divergences) 3 (A + B) = A + B (curl of sum = sum of curls) 4 (ΦA) = ( Φ) A + Φ( A) (product rule) 5 (ΦA) = ( Φ) A + Φ( A) (product rule) 6 ( Φ) = 0 (the curl of the gradient is zero) 7 ( A) = 0 (the divergence of the curl is zero) 8 ( A) = ( A) 2 A Dana Mackey (DIT) Maths 13 / 18
Properties of the gradient If Φ(x,y) = c is a curve in the plane then the vector Φ is perpendicular to the curve at each point. Example: If Φ(x,y) = x 2 + y 2 then the vector Φ = (2x,2y) is perpendicular to the circle x 2 + y 2 = 1 at each point (x,y). Similarly, if Φ(x,y,z) = c is a 3-dimensional surface, then Φ is perpendicular to the surface at each point. Dana Mackey (DIT) Maths 14 / 18
Directional derivative Let u be a vector (a direction) and Φ(x,y,z) be a scalar function. The directional derivative of Φ in the direction of u is defined as the dot product of the gradient vector and the direction vector D u (Φ) = Φ u Example: The directional derivative of Φ(x,y) = x 2 y + y 3 at the point (1,1) in the direction of u = i j is D u (Φ)(1,1) = Φ(1,1) u = (2i + 4j) (i j) = 2 1 + 4 ( 1) = 2. Dana Mackey (DIT) Maths 15 / 18
Direction of maximal increase: The gradient of a function Φ(x,y,z) at a point (x 0,y 0,z 0 ) defines the direction in which the function increases most rapidly at that point. The rate of maximum increase at that point is given by Φ(x 0,y 0,z 0 ). Similarly, the function decreases most rapidly in the direction of Φ. The rate of maximum decrease at that point is given by Φ(x 0,y 0,z 0 ). Any direction perpendicular to the gradient is a direction of zero change for the function. Example: Find the directions of maximal increase, maximal decrease and zero change for the function Φ(x,y) = x 2 + y 2 at (1,1). Dana Mackey (DIT) Maths 16 / 18
The physical meaning of curl and div The curl of a vector field A describes the rotation of the field A at a point. A vector field A for which A = 0 is called irrotational or conservative. The divergence of a vector field A measures how much it spreads out or diverges from a given point. A more rigorous interpretation as well as some physical examples will be given after we study line integrals. Dana Mackey (DIT) Maths 17 / 18
Example: Imagine that the xy-plane (or a small region around the origin) is filled with moving particles and that v(x,y) represents the velocity of the particle at (x,y). For each of the following examples, sketch the vector field v(x,y) and calculate its curl and divergence. 1 v(x,y) = yi + xj 2 v(x,y) = xi + yj 3 v(x,y) = xi yj Dana Mackey (DIT) Maths 18 / 18