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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Tangent Planes and Differentials">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 14:00:41 1997}
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\begin{document}


\section{Tangent Planes and Differentials\label%
{Tangent Planes and Differentials}}

Suppose a surface $S$ has equation $z=f(x,y)$, where $f$ has continuous
first partial derivatives, and let $p(x_{0},y_{0},z_{0})$ be a point on $S$.
Let $C_{1}$ and $C_{2}$ be the curves obtained by intersecting the vertical
planes $y=y_{0}$ and \ $x=x_{0}$ with the surface $S$. Then the point $P$
lies on both $C_{1}$ and $C_{2}$. Let $T_{1}$ and $T_{2}$ be the tangent
lines to the curves $C_{1}$ and $C_{2}$ at the point $P$. Then the tangent
plane to the surface $S$ at the point $P$ is defined to be the plane that
contains both of the tangent lines $T_{1}$ and $T_{2}$. \vspace{1pt}An
equation of the tangent plane to the surface $z=f(x,y)$ at the point $%
p(x_{0},y_{0},z_{0})$ is 
\[
z-z_{0}=f_{x}(x_{0,}y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0}) 
\]

\subsubsection{Differentials\label{Differentials}}

Consider a function of two variables , $z=f(x,y)$. If $x$ and $\ y$ are
given increments $\Delta x$ and $\Delta y$, then the corresponding \textbf{%
increment} of $z$ is 
\begin{eqnarray*}
\Delta z &=&f(x+\Delta x,y+\Delta y)-f(x,y) \\
&&
\end{eqnarray*}
Thus the increment $\Delta z$ represents the change in the value of $f$ when 
$(x,y)$ changes to $\left( x+\Delta x,y+\Delta y\right) $.

The \textbf{differentials} $dx$ and $dy$ are independent variables; that is,
they can be given any values. Then the \textbf{differential} $dz$, also
called the \textbf{total differential}, is defined by 
\[
dz=f_{x}(x,y)dx+f_{y}(x,y)dy=\dfrac{\partial z}{\partial x}dx+\dfrac{%
\partial z}{\partial y}dy 
\]

\begin{theorem}
Suppose that $f_{x}$ and $f_{y}$ exist on a rectangular region $R$ with
sides parallel to the axes and containing the points $(a,b)$ and ($a+\Delta
x,b+\Delta y)$. Suppose that $f_{x}$ and $f_{y}$ are continuous at the point 
$(a,b)$ and let 
\begin{eqnarray*}
\Delta z &=&f(a+\Delta x,b+\Delta y)-f(a,b) \\
&&
\end{eqnarray*}
Then 
\[
\Delta z=f_{x}(a,b)\Delta x+f_{y}(a,b)\Delta y+\varepsilon _{1}\Delta
x+\varepsilon _{2}\Delta y
\]
where $\varepsilon _{1}$ and $\varepsilon _{2}$ are functions of $\Delta x$
and $\Delta y$ that approach $0$ as $(\Delta x,\Delta y)\rightarrow (0,0)$.
\end{theorem}

\begin{definition}
If $z=f(x,y),$then $f$ is \textbf{differentiable} at $(a,b)$ if $\Delta z$
can be expressed in the form 
\[
\Delta z=f_{x}(a,b)\Delta x+f_{y}(a,b)\Delta y+\varepsilon _{1}\Delta
x+\varepsilon _{2}\Delta y
\]
where $\varepsilon _{1}$ and $\varepsilon _{2}\rightarrow 0$ as $(\Delta
x,\Delta y)\rightarrow (0,0)$.
\end{definition}

\subsubsection{Functions of Three or more variables\label%
{Functions of Three or more variables}}

Differentials and differentiability can be defined in a similar manner for
functions of more than two variables. For instance, if $w=f(x,y,x),$ then
the increment of $w$ is 
\begin{eqnarray*}
\Delta w &=&f(x+\Delta x,y+\Delta y,z+\Delta z)-f(x,y,z) \\
&&
\end{eqnarray*}
The differential $dw$ is defined in terms of the differentials $dx,dy,$and $%
dz$ of the independent variables by 
\[
dw=\dfrac{\partial w}{\partial x}dx+\dfrac{\partial w}{\partial y}dy+\dfrac{%
\partial w}{\partial z}dz
\]
If $dx=\Delta x,dy=\Delta y,$ and $dz=\Delta z$ are all small and $f$ has
continuous partial derivatives, then $dw$ can be used to approximate $\Delta
w.$

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Computing a derivative}{}{}{../../Help/DM7-3.tex#Derivative%
}}

{\small \hyperref{Function and expression names}{}{}{%
../../Help/DM5-1.tex#Function name}}

\noindent \hrulefill

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\noindent \hrulefill

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