MATH1011 - Lecture 7

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MATH1012 - Lecture 4
Skeleton Slides

Partial Derivatives

• A function can be partially derived by only deriving one value
  • e.g. for $f(x,y)$, the partial derivative of $f$ with respect to $x$ is written as $f_x(a,b)$
    • $y$ is considered a constant
  • Cannot use $f^{\prime }(a,b)$; must show which variable is being derived
    $\frac{\partial f}{\partial x}(a,b)=f_x(a,b)={\lim }_{h\to 0}\frac{f(a+h,b)-f(a,b)}{h}$
• $f_y(a,b)$ can be used for the partial derivative of $f$ with respect to $y$
  • In this case, $x$ is treated as a constant

Geometric Interpretation

• At a given point, the value of $f_x(x,y)$ = slope of the tangent line to the surface that is parallel to the $xz$−plane
• At a given point, the value of $f_y(x,y)$ = slope of the tangent line to the surface that is parallel to the $yz$−plane

• i.e. the two tangent vectors at a point are $[1,0,f_x],[0,1,f_y]$

Higher Derivatives

• Read Slides
  • Certain notation for higher derivatives
    • Dependent on the order of the partial derivative
• The order of the partial derivative can matter
  • $f_{xy}(x,y)$ may not equal $f_{yx}(x,y)$

Clairaut’s Theorem

• Let $f(x,y)$ be defined on an open disc $D$ of ${\mathbb{R}}^2$
• If the mixed derivative functions $f_{xy}$ and $f_{yx}$ are both defined and continuous on $D$, then $f_{xy}=f_{yx}$ on $D$

Tangent Planes

• In 3D space, the tangent to a point on a plane is another plane

  

• A plane can be constructed using the two tangent lines produced by the partial derivatives, $f_x(a,b)$ and $f_y(a,b)$, which are parallel to the tangent plane

• See Planes for more information:

  • Cross product of two parallel lines in the plane gives the normal
    • i.e. the normal at a point is equal to $[-f_x,-f_y,1]$
  • Plug into: $\mathbf{r}\cdot \mathbf{n}=\mathbf{a}\cdot \mathbf{n}$ and use the tangent point for $\mathbf{a}$

• Alternative formula: $z-f(a,b)=f_x(a,b)(x-a)+f_y(a,b)(y-b)$

  • Tangent plan at the point $(a,b,f(a,b))$
  • Typically written as $(z-f(a,b))-f_x(a,b)(x-a)-f_y(a,b)(y-b)=0$
  • Or: $-f_x(a,b)x-f_y(a,b)y+z=d$
    • Where $d$ is some constant

• For a surface $f(x,y,z)=c$, the tangent plane at a point $P=(x_0,y_0,z_0)$:
  $f_x(P)(x-x_0)+f_y(P)(y-y_0)+f_z(P)(z-z_0)=0$

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MATH1011 - Lecture 6
Annotated Slides