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
⑶ Operation 1: Vector Addition
① Methods : Parallelogram law, triangle law
② Property 1: Commutative law
③ Property 2: Associative law
⑷ Operation 2: Vector Subtraction
⑸ Operation 3: Scalar Multiplication of Vectors
⑹ Operation 4: Vector Resolution
① Definition : Decomposing each vector into component vectors along orthogonal coordinate directions
② Formulation
⑺ Operation 5: Scalar Product (Dot Product)
⑻ 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