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Chapter 1. Dynamics: Vectors and Scalars

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1. Representation of Physical Quantities : Scalar

2. Representation of Physical Quantities : Vector

3. Representation of Physical Quantities : Tensor

4. Two Approaches of Newtonian Mechanics



1. Representation of Physical Quantities : Scalar

⑴ Definition : Physical quantity with magnitude only

① Etymology : Latin word “scalae” meaning ‘steps’ or ‘ladder’ → scalaris → scala

② Examples : Time, length, area, volume, speed, mass, temperature, work, energy

⑵ Free in arithmetic operations



2. Representation of Physical Quantities : Vector

⑴ Definition : Physical quantity with both direction and magnitude

① Etymology : Latin word “vehere” meaning ‘to carry’ → vectus → vector

② Examples : Force, displacement, velocity, momentum, impulse, electric field, magnetic field

⑵ Notation of Vectors

① When represented with arrows : Arrow length represents vector magnitude. Arrow direction represents vector direction.

② When represented with symbols

③ Absolute value notation

Operation 1: Vector Addition

① Methods : Parallelogram law, triangle law

Property 1: Commutative law

Property 2: Associative law

Operation 2: Vector Subtraction

Operation 3: Scalar Multiplication of Vectors

Operation 4: Vector Resolution

① Definition : Decomposing each vector into component vectors along orthogonal coordinate directions

② Formulation

Operation 5: Scalar Product (Dot Product)

Operation 6: Vector Product (Cross Product)

① Definition of cross product : Operation satisfying three conditions for v, w ∈ ℝn:

○ v × w is a continuous function of the pair (v, w)

○ v × w is perpendicular to both v and w : (v × w) · v = (v × w) · w = 0

○ If v and w are linearly independent, then v × w ≠ 0

② In 3D vectors

○ Cross product of unit vectors : (Note) When operated in the order i → j → k, the minus sign does not appear

○ Fundamental formula : Cross product of unit vectors can be used for proof

○ Cross product direction : Direction of right-hand thumb when curling from A to B

○ Non-commutative property

○ Volume of parallelepiped : Volume of parallelepiped formed by vectors a, b, c is as follows

○ BAC-CAB rule

○ Proof is similar to scalar product proof : Start with a simple situation and generalize later

③ Multidimensional vectors

○ Satisfying the definition of cross product is only possible when n = 3 or n = 7

Reference paper : Massey, W. S. (1983). Cross products of vectors in higher dimensional Euclidean spaces. The American Mathematical Monthly, 90(10), 697-701.



3. Representation of Physical Quantities : Tensor



4. Two Approaches of Newtonian Mechanics

⑴ Newtonian Analysis : Analyzing dynamics through vector quantities like force and acceleration

⑵ Lagrangian Analysis : Analyzing dynamics through scalar quantities like energy

⑶ Due to the equivalence of differentiation and integration, there is equivalence between Newtonian and Lagrangian analyses



Input : 2016.06.26 21:05

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