Differential Calculus

• The derivative quantifies the instantaneous rate of change of a function. It is more formally defined as

  • We also sometimes notate this as if is explicitly clear.

• The partial derivative is a multivariable extension of the derivative that is calculated as the single variable derivative where all variables other than are held constant.

• The Leibniz Integral Rule for differentiation under the integral sign states that for an integral of the form

  where has a derivative expressible as

• The Chain Rule relates rates of changes between functions. It can be written as

  For the multivariate case, it is simply

• The gradient points to the direction of steepest ascent. It is perpendicular to the surface of the function.

  • Its components are the partial derivatives with respect to the coordinate system. That is

  • The dot product gives the directional derivative, the component of the gradient in the direction of

  • Let , then we say is a gradient field and is the potential function.

• The divergence of a continuously differentiable Vector Field is calculated as

  It is a measure of how much nearby vectors are diverging.

• The Laplacian of a function is defined as

• The curl of is defined as

  It measures the vorticity of the vectors, that is the tendency for nearby vectors to move in a circular direction.

• The divergence and curl are related as follows

  That is, the divergence of the curl is .

• Lagrange Multipliers work on the principle that given a function to be minimized subject to constraint , they must satisfy

  That is, their normal vectors at the critical point must be parallel. is the Lagrange Multiplier.

  • An alternative way to say this is to create the Lagrangian. That is, by creating a new function with which we minimize. This function is of the form
    The derivative of which with respect to and are minimized assuming we formulate that .

• The Jacobian Matrix of a function is whose first order partial derivatives are defined, is defined as an matrix where

  Or as a matrix

  Its Determinant called the Jacobian denotes the relative change in volume near a point . If it is positive it preserves orientation. Otherwise, it reverses orientation

• The Hessian Matrix of a function whose second order partial derivatives exist, is the square Matrix where

  Its determinant is called the Hessian.

  It is related to the Jacobian as followed:

Theorems §

• Let and be differentiable vector valued functions.

  The derivative of the dot product is given by

  • Proof: Use the definition of the dot product and apply the product rule

• Let and be differentiable vector valued functions.

  The derivative of their cross product is given by

  • Proof: Use the definition of the cross product and apply the product rule

• Let be a differentiable valued function such that each of its components are differentiable real functions.

  Let be a differentiable real-valued function.

  Then

Links §

• Integral Calculus
• Linear Algebra