MATH1011 - Lecture 6

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MATH1011 - Lecture 5
Skeleton Slides

Test of Differentiability

• If $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist on an open disc centred at $(a,b)$ and are continuous at $(a,b)$ then $f$ is differentiable at $(a,b)$
• And vice versa: if $f$ is differentiable at $(a,b)$, $f$ is continuous at $(a,b)$
  • Not every continuous function is differentiable!

The Jacobian Matrix

• Matrix containing all the partial derivatives of a multivariable function
  • Good luck, jk not that hard, look at slides

The Jacobian

• The Jacobian is the determinant of the Jacobian matrix when $m=n$
  • $m=n$ → equal rows and columns in the Jacobian matrix
  • Read Matrices for more info on determinant ($ad-bc$)

The Gradient

• The gradient vector of $f$ at $(a,b)$ is the vector:
  $\nabla f(a,b)=\text{grad}f(a,b)=(f_x(a,b),f_y(a,b))$

Chain Rule for Multivariable Functions

$\frac{d}{dt}f(x(t),y(t))=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}$

• This theorem applies for functions of any number of variables
  • e.g. $\frac{d}{dt}f(x_1(t),x_2(t),...,x_n(t))=\frac{\partial f}{\partial x_1}\frac{d{x}_1}{dt}+\frac{\partial f}{\partial x_2}\frac{d{x}_2}{dt}+...+\frac{\partial f}{\partial x_n}\frac{d{x}_n}{dt}$

General Chain Rule

• Read slides

Directional Derivative

• The directional derivative of $f$ at $c$ in direction $\vec{\mathbf{v}}$ is defined by $D_{v}f(c)={\lim }_{h\to 0}\frac{f(c+h\hat{\mathbf{v}})-f(c)}{h}=\nabla f(c)\cdot \hat{\mathbf{v}}$
  • $\hat{\mathbf{v}}=\frac{v}{|v|}$
  • Full definition on slides
• The initial function must be differentiable

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MATH1011 - Lecture 7
Annotated Slides