Neat Stuff from Vector Calculus

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Neat Stuff from Vector CalculusRelated Subjects

• Chris Hecker
• checker_at_d6.com
• definition six, inc. Maxis

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Related Subjects
scalar calculus
linear algebra
differential integral calculus
vector calculus
optimization constraints
differential geometry
classical mechanics, dynamics
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Intro Prerequisites

• this is a total hodge-podge, not a gentle
  introduction
• lack of rationale for caring about this stuff,
  trust me? )
• stuff that I found non-intuitive or hard to
  figure out
• tour of fairly basic examples building on
  themselves to get firm grounding and intuition in
  calculus concepts for the kinds of math games do
• touch on lots of different areas during tour,
  many sidestreets
• comfortable with algebra, linear algebra
• need to understand scalar calc, at least at the
  plugnchug level of differentiating functions
• f(x) ax2bxc f(x) 2ax b

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What is a Function?

• a function maps values from the domain to the
  range uniquely
• can be multivalued in range, but not domain

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What is a Derivative?

• derivative is another function that maps changes
  in the domain to changes in the range,
  normalized

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Derivatives are Linear

• the key insight of calculus the change is so
  small that you can ignore it anytime its
  multiplied by itself...so, you can treat any
  continuous function as linear if youre zoomed in
  far enough (to 1st order)
• continuity keeps us from dividing by zero
• normalization makes the numbers finite

boom!
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Random but Nifty Example of Ignoring
Infinitessimals(nothing up my sleeve!)

• prove infinitessimal rotations in 3D are vectors
  (add, commute, etc.)
• angular velocity is a vector, cross product
  differentiates rotating vectors

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Scalar Derivatives a line

• how does y (or f(x)) change for a change in x?
• for lines, the change (derivative) is constant
  everywhere
• drawn as red vector, but actually a scalar,
  slope

a
1
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Scalar Derivatives a curve

• for curves, the derivative is position dependent
• derivative is a function itself, mapping change
  in domain to change in range
• both direction and magnitude, but still a scalar

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The Shape of Matrix Operations

• vectors are columns of numbers in this talk, not
  rows
• we right-multiply matrices by column vectors v
  Mv
• matrix vector ops fit together nicely
• keeping the shapes right is the key to sanity
  with vector calc
• m by n n by p m by p
• this is why vvM makes no sense for column
  vectors
• either youre using row vectors, or youre
  confused
• either way, youre in for some pain when trying
  to do real math
• because all math books use columns for vectors
  and rows are special
• early computer graphics books got this backwards,
  and hosed everybody

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Shape of Derivativesdf needs to accept a Ddomain
linearly (ie. right-multiplied vector or scalar)
to produce a Drange

• scalar valued function of scalar
• yf(x) dy dfdx
• vector valued function of scalar
• vf(t) dv dfdt
• scalar valued function of vector
• zf(x,y)
• vector valued function of vector
• pf(u,v)

df
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Shape of Derivatives (cont.)

• scalar valued function of vector expands to row
  vector
• the resulting range value depends on all the
  domain values
• the differential needs a slot for a delta/change
  in each domain dimension...so it must be a row
  vector, theres no T or
• zf(x,y)
• dzdfdx
• vector valued function of vector expands to
  matrix
• pf(u,v)

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Derivatives a parametric functiona vector
function of a scalar

• for a change in the parameter (domain), how does
  the function (range) change?
• in this case, differential is a vector

q
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Derivatives a scalar function of a
vectorheight field

• differential is not in range of function

column vector
row vector
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Derivatives a scalar function of a vector

• view height field as implicit surface in 3d
• write g(x,y,z) gt 0 above surface
• differential is surface normal (not unit)

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Derivatives a scalar function of a vector

• example plane

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Derivatives a scalar function of a vector

• example sphere

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A Surface Normal is Not a Vector!this is why you
need to keep shapes distinct

• vector is a difference between two points
• normal is really a mapping from a vector to a
  scalar

points vectors transform like this
normals transform like this
n
b
v
a
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Normals and Vectors Example

• scaling an ellipse

x values for y0
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Derivative of Vector Mappings

• barycentric coordinates in 2D triangle
• vector function of a vector
• if square, can invert Jacobian to find du,dv
  given dp
• function is linear in this case, but works
  generally
• determinant of Jacobian is how areas distort
  under function

p2
p
e2
p1
e1
p0
Jacobian matrix
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Barycentric Coordinates in 3D Triangle

• now jacobian is 3x2
• can still find du,dv from dp with least squares

p2
p
same as projecting dp down into triangle
e2
p1
e1
p0
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Implicit Functions

• equalities (constraints) subtract off DOFs
• explicit to implicit is easy z f(x,y)
  g(x,y,z) z f(x,y) 0
• implicit to explicit is hard f(x,y) 0 y
  f(x)
• solving nonlinear equations, sometimes multiple
  or no solutions
• but, inverting it differentially is easy because
  of linearization

f(x,y) 0
shape/rank of jacobian will tell you how
constrained you are as well
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Product Rule for Vector Derivatives

• for scalar multiplication
• differential of a scalar function of a scalar is
  a scalar

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Product Rule for Vector Derivatives

• for scalar multiplication
• differential of a scalar function of a scalar is
  a scalar
• for dot product its a little wackier
• if vectors in dot are functions of scalars, its
  the same

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Product Rule for Vector Derivatives

• for scalar multiplication
• differential of a scalar function of a scalar is
  a scalar
• for dot product its a little wackier
• if vectors in dot are functions of scalars, its
  the same
• if vectors in dot are functions of vectors, need
  to watch shape!
• we know result must be row

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Product Rule for Vector Derivatives

• for scalar multiplication
• differential of a scalar function of a scalar is
  a scalar
• for dot product its a little wackier
• if vectors in dot are functions of scalars, its
  the same
• if vectors in dot are functions of vectors, need
  to watch shape!
• we know result must be row

use transpose picture to reason about it
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Product Rule for Vector Derivatives

• for scalar multiplication
• differential of a scalar function of a scalar is
  a scalar
• for dot product its a little wackier
• if vectors in dot are functions of scalars, its
  the same
• if vectors in dot are functions of vectors, need
  to watch shape!
• we know result must be row

use transpose picture to reason about it
dont want to use tensors, so we pull a fast one
with the commutativity of dot (abba)
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Dot Product Derivative Example

• derivative of squared length of vector

makes intuitive sense if dp is orthogonal,
length doesnt change
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Cross Product Works the Same Way

• if cross is a vector function of vectors...
• result must be a matrix
• use the skew symmetric picture of cross product

same problem with tensors, so we pull the same
(skew-)commutativity trick
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Cross Product Derivative Example

• differentiate a cross product of moving vector
  with constant vector

n
dr
r
c
dc
again, makes intuitive sense dc changes
orthogonally to dr changes
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References

• Advanced Calculus of Several Variables
• Edwards, Dover
• Calculus and Analytic Geometry
• Thomas Finney
• Classical Mechanics
• Goldstein