Dr. Wang Xingbo Fall ， 2005 Mathematical & Mechanical Method in Mechanical Engineering

1.Vectors 2.Products of Two Vectors 3.Vector Calculus 4.Fields 5.Applications of Gradient, Divergence and Curl Mathematical & Mechanical Method in Mechanical Engineering Vector Algebra

Quantities that have both magnitude and direction; the magnitude can stretch or shrink, and the direction can reverse. In a 3-dimmensional space, a vector X=(x 1, x 2, x 3 ) has three components x 1,x 2, x 3. Mathematical & Mechanical Method in Mechanical Engineering Vectors

Vectors X=(x 1, x 2, x 3 ), Y=(y 1, y 2, y 3 ) Scalar multiplication:2X = (2x 1, 2x 2, 2x 3 ) Addition:X + Y = (x 1 + y 1, x 2 + y 2, x 3 + y 3 ) The zero vector:0 = (0,0,0) The zero vector:0 = (0,0,0) The subtraction: X - Y = (x 1 - y 1,x 2 - y 2,x 3 - y 3 ) Mathematical & Mechanical Method in Mechanical Engineering Algebraic properties

Length of X = (x 1, x 2, x 3 ) is calculated by: A unit vector in the direction of X is : Mathematical & Mechanical Method in Mechanical Engineering Length (magnitude) of a vector

Proj u A = (|A| cos  )u ( |u| = 1) Mathematical & Mechanical Method in Mechanical Engineering Projection of a Vector

1.Inner Product,doc product,scalar product 2.Vector Product,cross product 3.Without extension Mathematical & Mechanical Method in Mechanical Engineering Products of Two Vectors

A=(a 1, a 2, a 3 ), B=(b 1, b 2, b 3 ) Mathematical & Mechanical Method in Mechanical Engineering Inner Product

1.Non-negative law 2. Commutative law: 3.Distributive law: Mathematical & Mechanical Method in Mechanical Engineering Properties of Scalar Product

1.Cross product of two vectors A and B is another vector C that is orthogonal to both A and B 2. C = A × B 3.|C| = |A||B||sin  | Mathematical & Mechanical Method in Mechanical Engineering Vector Product

1.The length of C is the area of the parallelogram spanned by A and B 2. The direction of C is perpendicular to the plane formed by A and B; and the three vectors A, B, and C follow the right-hand rule. Mathematical & mechanical Method in Mechanical Engineering Geometric Meanings of Cross Product

1. A×B = -B ×A, 2. A ×(B + C) = A ×B +A ×C, 3. A||B is the same as A ×B = 0 Mathematical & mechanical Method in Mechanical Engineering Properties of Cross Product

1.i 1 × i 1 = 0, i 2 × i 2 = 0, i 3 × i 3 = 0, 2.i 1 × i 2 = i 3, i 2 × i 3 = i 1, i 3 × i 1 = i 2 Mathematical & mechanical Method in Mechanical Engineering Three Basis Vectors A = a 1 i 1 + a 2 i 2 + a 3 i 3, B = b 1 i 1 + b 2 i 2 + b 3 i 3

1.(A × B) × C = B(A·C) -A(B·C) 2.A × (B × C) = B(A·C) - C(A·B) Mathematical & mechanical Method in Mechanical Engineering Product of Three Vectors A = a 1 i 1 + a 2 i 2 + a 3 i 3, B = b 1 i 1 + b 2 i 2 + b 3 i 3, C = c 1 i 1 + c 2 i 2 + c 3 i 3

1. A · (B×C) = (A×B) · C = (C×A) · B 2. (A× B) · (C×D) = (A · C)(B · D) - (A · D)(B · C) 3. (A× B) · (A×C)= B · C - (A · C) (A · B) 4.(A×B) ·(C×D) + (B×C) ·(A×D) + (C×A) ·(B×D) = A×(B×C) + B×(C×A) + C×(A×B) = 0 6. (A×B) ×(C×D) = C(A · (B×D)) - D(A · (B ×C)) = B(A·(C×D))-A(B·(C×D)) Mathematical & mechanical Method in Mechanical Engineering Other Useful Formula for Vector Products

For any scalar t, a function f(t) is called a vector function or a variable vector if there exists a vector corresponding with f(t). A(t) = (cos t, sin t, 0) (- ∞ < t < ∞ ) Mathematical & mechanical Method in Mechanical Engineering Vector Calculus

Mathematical & mechanical Method in Mechanical Engineering The Derivatives of a Vector Function A(t) = (A 1 (t),A 2 (t),A 3 (t)) = A 1 (t)i 1 + A 2 (t)i 2 + A 3 (t)i 3

Mathematical & mechanical Method in Mechanical Engineering Properties of Vector Derivative velocity acceleration

Mathematical & mechanical Method in Mechanical Engineering Properties of Vector Derivative

A(t) = (A 1 (t),A 2 (t),A 3 (t)) = A 1 (t)i 1 + A 2 (t)i 2 + A 3 (t)i 3 Mathematical & mechanical Method in Mechanical Engineering The Integral of a Vector Function

Suppose Ω be a subspace, P be any point in Ω, if there exists a function u related with a quantity of specific property U at each point P , namely, Ω is said to be a field of U if Mathematical & mechanical Method in Mechanical Engineering Fields where symbol means “subordinate to”

1. Temperature in a volume of material is a temperature field since there is a temperature value at each point of the volume. 2. Water Velocity in a tube forms a velocity field because there is a velocity at each point of water in the tube. 3. Gravity around the earth forms a field of gravity 4. There is a magnetic field around the earth because there is a vector of magnetism at each point inside and outside the earth. Mathematical & mechanical Method in Mechanical Engineering Example of fields

A real function of vector r in a domain is called a scalar field. Pressure function p(r) and the temperature function T(r) in a domain D are examples of scalar fields. Mathematical & mechanical Method in Mechanical Engineering Scalar Fields A scalar field can be intuitionistically described by level surfaces

Directional Derivative Mathematical & Mechanical Method in Mechanical Engineering Directive derivatives and gradient Directive derivatives and gradient Where l is a unit vector

Gradient Mathematical & Mechanical Method in Mechanical Engineering Directive derivatives and gradient Directive derivatives and gradient It can be shown if l is a unit vector

Properties 1.The gradient gives the direction for most rapid increase. 2.The gradient is a normal to the level surfaces. 3.Critical points of f are such that  =0 at these points Mathematical & Mechanical Method in Mechanical Engineering Directive derivatives and gradient Directive derivatives and gradient

Mathematical & Mechanical Method in Mechanical Engineering Operational rules for gradient Operational rules for gradient

Two important concepts about a vector field are flux,divergence, circulation and curl Mathematical & Mechanical Method in Mechanical Engineering Vector Fields Vector Fields A vector field can be intuitionistically described by vector curve tangent at each point to the vector that is produced by the field

The Flux is the rate at which some- thing flows through a surface. Mathematical & Mechanical Method in Mechanical Engineering Flux Flux Let A= A (M) be a vector field, S be an orientated surface, A n be normal component of the vector A over the surface S

A(r)=(A 1 (x 1, x 2, x 3 ),A 2 (x 1, x 2, x 3 ),A 3 (x 1, x 2, x 3 )) in Cartesian coordinate system Mathematical & Mechanical Method in Mechanical Engineering Flux Flux

Rate of flux to volume. In physics called density. Mathematical & Mechanical Method in Mechanical Engineering Divergence Divergence

A(r)=(A 1 (x 1, x 2, x 3 ),A 2 (x 1, x 2, x 3 ),A 3 (x 1, x 2, x 3 )) In Cartesian coordinate system Mathematical & Mechanical Method in Mechanical Engineering Divergence Divergence

Lagrangian Operator Mathematical & Mechanical Method in Mechanical Engineering Divergence Divergence

Mathematical & Mechanical Method in Mechanical Engineering Operational rules for divergence Operational rules for divergence

Circulation is the amount of something through a close curve Mathematical & Mechanical Method in Mechanical Engineering Circulation Circulation A(  ) be a vector field, l be a orientated close curve

A(r)=(A 1 (x 1, x 2, x 3 ),A 2 (x 1, x 2, x 3 ),A 3 (x 1, x 2, x 3 )) l be a orientated close curve Mathematical & Mechanical Method in Mechanical Engineering Circulation Circulation

A(r)=(A 1 (x 1, x 2, x 3 ),A 2 (x 1, x 2, x 3 ),A 3 (x 1, x 2, x 3 )) Mathematical & Mechanical Method in Mechanical Engineering The Curl of a Vector Field The Curl of a Vector Field

Makes circulation density maximal at a point along the curl. Makes circulation density maximal at a point along the curl. Mathematical & Mechanical Method in Mechanical Engineering The Curl of a Vector Field The Curl of a Vector Field

Mathematical & Mechanical Method in Mechanical Engineering Operational rules for Rotation (Curl) Operational rules for Rotation (Curl)

Potential Field Potential Field A=grad  Tube Field divA =0 Harmonic Field divA =0, rotA=0 Mathematical & Mechanical Method in Mechanical Engineering Several Important Fields Several Important Fields

Mathematical & Mechanical Method in Mechanical Engineering Summary Summary

Class is Over! See you Friday Evening! Mathematical & Mechanical Method in Mechanical Engineering 21:30,1,Dec, :30,1,Dec,2005