Chapter Review

Definitions:

• Function \(z=f(x,y)\) of two variables (p. 810)
• Function \(w=f(x,y,z)\) of three variables (p. 810)
• Contour line: A curve that is the intersection of a surface \(z=f(x,y)\) and the plane \( z=c,\) where \(c\) is a constant (p. 812)
• Level curve: the projection of a contour line onto the \(xy\)-plane (p. 813)

Definitions:

• A \(\delta \)-neighborhood of \(P_{0}\): the set of all points \(P\) that lie within a distance \(\delta \) of \(P_{0}\) (p. 819)

• In the plane, a \(\delta \)-neighborhood of \(P_{0}\) is called a disk of radius \(\delta\) centered at \(P_{0}.\)
• In space, a \(\delta\)-neighborhood of \(P_{0}\) is called a ball of radius \(\delta\) centered at \(P_{0}.\)

• Limit of a function \(f\) of two variables (p. 819)
• Limit of a function \(f\) of three variables (p. 820)
• Properties of limits (p. 821)
• Interior point, boundary point, open set, closed set (p. 825)
• Continuity at a point (p. 825)
• A function \(f\) is continuous if it is continuous at every point in its domain (p. 826)
• Properties of continuous functions (p. 826)

• First-order partial derivatives of a function \(z=f(x,y) \): \begin{eqnarray*} f_{x}(x,y) &=& \dfrac{\partial f}{\partial x}=\dfrac{\partial z}{\partial x}=\lim\limits_{\Delta x\rightarrow 0} \dfrac{f( x+\Delta x,y) -f(x,y) }{\Delta x}\\[4pt] f_{y}(x,y) &=& \dfrac{\partial f}{\partial y}=\dfrac{\partial z}{\partial y}=\lim\limits_{\Delta y\rightarrow 0} \dfrac{f( x,y+\Delta y) -f(x,y) }{\Delta y} \end{eqnarray*} provided the limits exist (p. 830)
• Mixed partials: \(\dfrac{\partial ^{2}z}{\partial x~\partial y} =f_{yx}(x,y) \) and \(\ \dfrac{\partial ^{2}z}{\partial y~\partial x}=f_{xy}(x, y)\) (p. 834)
• Equality of mixed partials (Clairaut’s Theorem) (p. 835)

• Change in \(z\): \(\Delta z=f(x_{0}+\Delta x, y_{0}+\Delta y)-f(x_{0}, y_{0})\) (p. 841)
• Differentiability of \(z=f(x,y)\) at a point \((x_{0},y_{0})\) (p. 842)
• Differentials: \(dx=\Delta x,\) \(dy=\Delta y,\) \(dz=f_{x}(x_{0},y_{0}) \,dx+f_{y}(x_{0},y_{0})\,dy\) (p. 843)

Summary: (p. 846) If \(z=f(x,y) ,\)

• The continuity of the partial derivatives \(f_{x}\) and \(f_{y}\) implies the differentiability of \(f\).
• The differentiability of \(f\) implies the continuity of \(f\).
• The continuity of the partial derivatives \(f_{x}\) and \(f_{y}\) implies the continuity of \(f\).
• The existence of the partial derivatives \(f_{x}\) and \(f_{y}\) does not necessarily mean \(f\) is differentiable.
• The existence of the partial derivatives \(f_{x}\) and \(f_{y}\) does not necessarily mean \(f\) is continuous.

• Chain Rule I: \(\dfrac{dz}{dt}=\dfrac{\partial z}{\partial x}\dfrac{dx }{dt}+\dfrac{\partial z}{\partial y}\dfrac{dy}{dt}\) (p. 849)
• Chain Rule II: \[ \begin{eqnarray*} \dfrac{\partial z}{\partial u}&=&\dfrac{\partial z}{ \partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y} \dfrac{\partial y}{\partial u} \\ \dfrac{\partial z}{\partial v}&=&\dfrac{ \partial z}{\partial x}\dfrac{\partial x}{\partial v}+\dfrac{\partial z}{ \partial y}\dfrac{\partial y}{\partial v} \end{eqnarray*} \] (p. 851)
• Implicit differentiation formula I: If \(F(x,y) =0,\) \( \dfrac{dy}{dx}=-\dfrac{F_{x} (x,y)}{F_{y}(x,y)}\) provided \(F_{y}\ne 0.\) (p. 854)
• Implicit differentiation formula II: If \(F(x,y,z) =0,\) \(\dfrac{\partial z}{\partial x}=-\dfrac{F_{x}(x,y,z)}{F_{z}(x,y,z)}\) and \( \dfrac{\partial z}{\partial y}=-\dfrac{F_{y}(x,y,z)}{F_{z}(x,y,z)}\) provided \(F_{z}\ne 0.\) (p. 855)