Vector Calculus Examples of Vector Quantities Vector is

Vector Calculus

Examples of Vector Quantities

Vector is Position Free

Vectors in a Coordinate Plane Position Vector

Sum & Difference of Position Vectors

Properties of Vectors 1. 2. 3. 4. 5. 6. 7. 8. 9. a+b=b+a (Commutative Law) a + (b + c ) = (a + b) + c (Associative Law) a+0=a (Additive Identity) a + (-a) = 0 (Additive Inverse) k(a+b) = ka + kb k : scalar (k 1 + k 2)a = k 1 a + k 2 a k 1(k 2 a) = (k 1 k 2)a 1 a = a 0 a = 0 (Zero Vector)

Magnitude(Length, Norm) of Vectors P(a 1, a 2) u Unit Vector u = (1/||a||)a

Cartesian Coordinates

i, j, k : Basis of R 3

Component Definitions in 3 -Space

Scalar Product(내적)

Scalar Product(내적)

Criteria for Orthogonal Vectors

Scalar Product(내적)

Scalar Product(내적)

Scalar Product(내적) Properties of the Scalar Product 1. a∙b = 0 if a = 0 or b = 0 2. a∙b = b∙a (Commutative Law) 3. a∙(b + c) = a∙b + a∙c (Distributive Law) 4. a∙(kb) = (ka)∙b = k(a∙b) k : Scalar 5. a∙a ≥ 0 6. a∙a = ||a||2

Scalar Product(내적) Definition b a a : b = acosθ : bcosθ 0≤θ≤π

Distributive Law b+c b a c a∙(b + c) = a∙b + a∙c

Vector(Cross ) Product Torque, Moment τ= r x F

Vector(Cross ) Product 0≤θ≤π

Vector(Cross ) Product

Area

Volume a∙(b x c) = 0 iff a, b, c are coplaner

Criteria for Parallel Vectors

Parallel Vectors ixj=k jxk=i kxi=j j x i = -k k x j = -i i x k = -j a = a 1 i + a 2 j + a 3 k b = b 1 i + b 2 j + b 3 k

Volume a∙(b x c) = 0 iff a, b, c are coplaner

Vector(Cross ) Product Properties of the Cross product 1. a x b = 0 if a = 0 or b = 0 2. a x b = - b x a 3. a x (b + c) = (a x b) + (a x c) 4. (a + b) x c = (a x c) + (b x c) 5. a x (kb) = (ka) x b = k(a x b) 6. a x a = 0 7. a∙(a x b) = 0 8. b∙(a x b) = 0

Distributive Law c+b b b c c a ax(b + c) = axb + axc

Distributive Law (a + b) x c = - c x (a + b) =-cxa-cxb =axc+bxc

Motion on a Curve Motion : 시간에 따른 위치의 변화

Motion on a Curve

벡터 함수(Vector Valued Functions) Parametric Curve in Space Vector Valued Function A vector r whose components are functions of a parameter t

Ellipse in a Plane Parametric Curve in Space

Circle in Space Parametric Curve in Space

Circular Helix(원 나선)

Derivative of Vector Valued Functions Tangent Vector Derivative of a Vector Valued Function r

Derivative of Vector Valued Functions Derivative of a Vector Valued Function r

Derivative of Vector Valued Functions

Rules of Differenciation : diffrentiable vector valued functions : diffrentiable scalar function

Integrals of Vector Valued Functions

Velocity, Acceleration Tangent Vector Derivative of a Vector Valued Function r Velocity : Acceleration : Speed :

Velocity, Acceleration Velocity : Acceleration : Parametric Curve in Space

Normal Unit Tangent Vector are Orthogonal

Acceleration

스칼라 장(Scalar Fields)

Partial Derivative differential 기울기 벡터 unit direction i, j

Scalar장의 도 함수(Gradient, Derivative)

Scalar장의 도 함수(Gradient) 3 Dim C의 최대 증가 방향 Level Surface r(t) : Curves on Level Surface (Movements) Gradient f : Vector Level Surface의 공간적인 최대 변화 벡터 는 P점에서 Level Surface와 수직이다.

Gravitational Field

Scalar장의 도함수(Gradient)

Scalar장의 도함수(Gradient)

Surface Normal Vector C의 최대 증가 방향 Level Surface Tangent Plane : Vector on Tangent Plane Eq. of Tangent Plane

Surface Normal Vector

벡터 장(Vector Fields) Vector Fields : 정해진 공간 상의 각 지점마다 Vetor 물리량을 갖는 장 § 열의 흐름 § 유체의 흐름(Fluid Flow) § 전. 자기장(Elecro. Magnetic Fields) § 중력장(Gravitational Fields)

벡터 장(Vector Fields)

벡터 장(Vector Fields)

Vector Field

Vector Algebra

Vector Derivative Operator : : 장의 도함수

Vector(Cross ) Product Turn the Bolt about Vertical Axis Torque, Moment τ= r x F

Curl(Decomposition of Rotational Derivative) (참고)

Curl

Curl

Curl

Curl

Divergence Flux of Vector Field F : 단위시간 당 면적 를 통해 흐르는 유량

Divergence Flux of F through Total Flux of F through and

Divergence Total Flux of F through yz plane Total Flux of F through xy plane Net Flux of F through all plane

Divergence Total Flux of F Unit Volumn 한 점에서 유체의 밀도 변화율의 척도

Divergence

Divergence(Sum of Partial Derivative)

Line Integral

Geometric Interpretation In Case of 2 Variables

Line Integral in the Plane Line Integral of G along C from A to B with respect to x Line Integral of G along C from A to B with respect to y Line Integral of G along C from A to B with respect to arc length

Line Integral in the Plane

Line Integral in Space

Line Integral Evaluation (Curve Defined by an Explicit Function)

Work Done by a Force Work done by F along C Unit Tangent to C

Circulation Line Integral of a Vector Field around Simple Closed Curve Circulation F가 유체의 속도장인 경우 C를 회전하 면서 C를 돌리는 양의 척도 >0 : CCW <0 : CW