Applied Calculus of Variations for Engineers, 2nd Edition

List of notations

Notation Meaning

f(x) Function of one variable

f(x, y),F(x, y) Function of two variables

r Radius of curvature

g Acceleration of gravity

r

Vector in Cartesian coordinates

r

x

First partial derivative with respect to x

r

y

First partial derivative with respect to y

˙r

First parametric derivative

¨r

Second parametric derivative

p Pressure

s Arc length

s(t) One-dimensional spline function

s(u, v) Two-dimensional spline function

y



(x),f



(x)Firstderivative

y



(x),f



(x) Second derivative

n

Normal vector

t

Tangent vector

b

Bi-normal vector

∇ Gradient operator

Δ Laplace operator

κ Curvature

κ

g

Geodesic curvature

κ

n

Normal curvature

κ

m

Mean curvature

δI Variation of integral functional

Γ

k

ij

Christoffel symbols

σ Stress

 Strain

ν Poisson’s ratio

λ Eigenvalue

ρ Material density

[B] Shape function derivative matrix

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xx Contents

B

i,k

(t) B-spline basis function

E,F,G First fundamental quantities

E Young’s modulus

[D] Elasticity matrix

F Active force

G Green’s function

E

s

Strain energy

E

k

Kinetic energy

E

p

Potential energy

I() Integral functional

I Moment of inertia

[J] Jacobian matrix

M Momentum

[K]Stiffnessmatrix

[M] Mass matrix

[N] Shape function matrix

Q Heat source

S Surface area

T Surface (traction) force

[T ] Temperature matrix

V Volume

W

e

External work