Lecture 1 Flashcards
(15 cards)
What is the difference between a scalar and a vector?
A scalar is a single quantity with only magnitude
A vector has both magnitude and direction
What is the dot product, and what does it represent?
dot product of two vectors is given by.
a⋅b=∣a∣∣b∣cosθ
In index notation
a⋅b=a_ib_i
Represents pojection of one vector onto another
What is the cross product, and what does it represent?
cross product produces a vector perpendicular to both input vectors:
a×b=∣a∣∣b∣sinθn
In Index notation using Levi-Civita symbol
(a×b)_i=ϵ_ijka_jb_k
What is the Einstein summation convention?
If an index appears twice, summation is implicitly undersood.
a_ib_i=∑a_ib_i
How does the Kronecker delta δ_ij function?
Kronecker delta acts as an identity matrix:
𝛿𝑖𝑗={1,if𝑖=𝑗}
𝛿𝑖𝑗={0,if𝑖≠𝑗}
reduces expression by selecting terms
What is the Levi-Civita symbol ϵijk?
Totally antisymmetric tensor
ϵijk={+1, even permutation}
ϵijk={-1, odd permutation}
ϵijk={0, if any indices are repeated}
Used to define cross products and determinants
What does the gradient of a scalar function represent?
Gradient points in the direction of maximum increase:
∇ϕ= ∂ϕ/∂x e_x+∂ϕ/∂y e_y+∂ϕ/∂z e_z
Describes changes in fields like temperature, pressure, or potential.
What is the divergence of a vector field?
Measures how much a vector spreads out or converges:
∇⋅F=∂F1/∂x+∂F2/∂y+∂F3/∂z
∇⋅F>0 → source (expanding field)
∇⋅F<0 → sink (contracting field)
What is the curl of a vector field?
Measures how much a field is rotating.
∇xF
What is a second-order tensor, and how is it different from a vector?
A vector has one direction:
𝑣=𝑣_𝑖𝑒_𝑖
A tensor has two directions:
𝐴=𝐴_𝑖𝑗𝑒_𝑖⊗𝑒𝑗
Example: Stress tensor, which describes forces in multiple directions.
When is a tensor symmetric?
A tensor is symmetric if:
𝐴_𝑖𝑗=𝐴_𝑗𝑖
When is a tensor anti-symmetric?
tensor is anti-symmetric if:
𝐴𝑖𝑗=−𝐴𝑗i
Why is 𝐵⋅𝐵 always symmetric if 𝐵 is anti-symmetric?
B^T=−B
The product:
(𝐵⋅𝐵)^𝑇=𝐵^𝑇⋅𝐵^𝑇=(−𝐵)⋅(−𝐵)=𝐵⋅𝐵
Thus, 𝐵⋅𝐵 is always symmetric.
What is the Divergence Theorem?
Converts a volume integral to a surface integral:
∫_𝑉(∇⋅𝐹)𝑑𝑉=∮_𝑆𝐹⋅𝑑𝑆
What is Stokes’ Theorem?
Converts a surface integral of curl into a line integral:
∮_𝐶𝐹⋅𝑑𝑟=∫_𝑆(∇×𝐹)⋅𝑑𝑆
