Multivariable Calculus Jeremy Irvin and Daniel Spokoyny - - PowerPoint PPT Presentation

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multivariable calculus
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Multivariable Calculus Jeremy Irvin and Daniel Spokoyny - - PowerPoint PPT Presentation

Jul 19, 2023 232 likes •460 views

Multivariable Calculus Jeremy Irvin and Daniel Spokoyny Derivative Let be open. A function is differentiable at if exists. We call this value f(x) . Intuitively, a function is differentiable if it is locally

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Multivariable Calculus

Jeremy Irvin and Daniel Spokoyny

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Derivative

  • Let be open. A function

is differentiable at if

  • exists. We call this value f’(x).
  • Intuitively, a function is differentiable if it

is locally approximated by a line.

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Derivative

  • If f is differentiable, we can rewrite this as
  • Or equivalently,
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Derivative

  • Now let be open, and .

Then f is differentiable at if such that

  • Intuitively, f is differentiable if it is locally

approximated by a linear function.

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Partial Derivative

  • Let . The jth partial derivative of f

at is provided this limit exists.

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Total and Partial Derivative

  • Now let and . We can

write

  • And if and , writing the

component of x out explicitly,

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Total and Partial Derivative

  • So we can define
  • Because f’(x) is linear, it can be represented

as a matrix, call it . In fact,

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Total and Partial Derivative

  • If f is real-valued ( ), then the

matrix representation of f’(x) is called the gradient of f at x, denoted

  • Think of the gradient as a direction in

which the parameters move so that the function f increases the fastest.

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Convexity

  • Let . A function is convex if

for all ,

  • This means that the line between any two

points on the graph of f lies above the graph

  • f f (see blackboard).
  • Convex functions have a single global
  • ptimum.
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Lagrange Multipliers

Jeremy Irvin and Daniel Spokoyny

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Lagrange Multipliers

  • Suppose is continuously

differentiable, and let M be the set of points such that and . If the differentiable function attains its maximum or minimum on M at the point , then is called the “Lagrange multiplier”.

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Lagrange Multiplier Example

  • Find the rectangular box of volume 1000

which has the least total surface area A.

  • Let and

.

  • We want to minimize f on the set of points

which satisfy .

  • Sounds like Lagrange Multipliers!
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Lagrange Multiplier Example

  • We want to solve
  • It is easily seen that the unique solution to

this set of equations is x=y=z=10.

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Generalized Lagrange Multipliers

  • Informally, given some constraints

, and denoting M the set

  • f points which satisfy them, if (under some

conditions on these constraints) the differentiable function attains a local maximum or minimum on M at , then

  • So to find points which optimize f given

some constraints, simply solve the set of equations above.

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Matrix Calculus

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Matrix Gradient

  • Let (input matrix, output real

value). The gradient of f with respect to some input is the matrix of partial derivatives:

  • More compactly,
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Matrix Derivative Properties

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Hessian

  • Suppose . The Hessian matrix

with respect to x is the n x n matrix of partial derivatives:

  • More compactly,
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Gradients/ Hessians of Quadratic/ Linear Functions

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What Just Happened?

  • Multivariable Derivative
  • Convexity
  • Lagrange Multipliers
  • Matrix Calculus
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Done with math!! (kinda)