$\grad F$ $B$O%l%Y%k%;%C%H$NK!@~%Y%/%H%k(B

$B0J2 $B$O(B $\R^n$ $B$N3+=89g!"(B $F\colon\Omega\to\R$ $B$O(B $C^1$ $B5i$H$9$k!#(B $c\in\R$ $B$KBP$7$F!"(B

$$L_c:=\left\{x\in\Omega; F(x)=c\right\}$$

$B$H$*$-!"(B$F$ $B$N%l%Y%k(B $c$ $B$N(B$B%l%Y%k!&%;%C%H(B$B$H8F$V!#(B

$L_c=\emptyset$ $B$H$J$k$3$H$b$"$j$&$k$,!"(B $a\in\Omega$ $B$N$H$-!"(B$c:=F(a)$ $B$H$*$/$H!"(B$a\in L_c$ $B$,@.$jN)$D$N$G!"(B $L_c\ne\emptyset$ $B$G$"$k!#(B

$B $\nabla F(a)$ $B$O!"(B $L_c$ $B$NE@(B $a$ $B$K$*$1$kK!@~%Y%/%H%k(B$B$H8F$V$K$U$5$o$7$$$b$N$G$"$k!#(B $B

$$\lim_{x\to a}\frac{F(x)-F(a)-F'(a)(x-a)}{\left\Vert x-a\right\Vert}=0$$

$B$,@.$jN)$D$,!"(B$c:=F(a)$ $B$H$9$k$H$-!"(B $\forall x\in L_c$ $B$KBP$7$F(B $F(x)=c$ $B$G$"$k$3$H$H!"(B $F'(a)(x-a)=\left(\nabla F(a),x-a\right)$ $B$G$"$k$3$H$+$i!"(B

$$\frac{F(x)-F(a)-F'(a)(x-a)}{\left\Vert x-a\right\Vert} =\frac{c-c-\left(\nabla F(a),x-a\right)}{\left\Vert x-a\right\Vert} =-\frac{\left(\nabla F(a),x-a\right)}{\left\Vert x-a\right\Vert}$$

$$=-\left(\nabla F(a),\frac{x-a}{\left\Vert x-a\right\Vert}\right)$$

$B$H$J$k$N$G!"(B

$$\lim_{x\in L_c\atop x\to a} \left(\nabla F(a),\dfrac{x-a}{\left\Vert x-a\right\Vert}\right)=0.$$

($B$3$N$"$?$j$N5DO@$O!V9M;!!W$G$"$j!"(B $B!V>ZL@!W$G$O$J$$!#LUL\E*$K?.$89~$^$l$F$b:$$k$N$G!"(B $B>/$7aS$r;XE&$7$F$*$/$H!"(B $L_c$ $B$,$I$&$$$&=89g$G$"$k$+!"(B $B0lHL$K$ONI$/J,$+$i$J$$!#$=$b$=$b(B $L_c$ $B$r$?$I$C$F!"(B $a$ $B$K6a$E$/$3$H$,=PMh$k$N$+$bJ,$+$i$J$$!#(B $B8e$G!V1"4X?tDjM}!W$H$$$&DjM}$r3X$V$H$3$N$"$?$j$,2r7h$5$l$k!#(B)

$\nabla F(a)\ne 0$, $x\ne a$ $B$G$"$k$H$-!"(B $\nabla F(a)$ $B$H(B $x-a$ $B$N$J$93Q(B $\theta(x)\in[0,\pi]$ $B$,(B

$$\left(\nabla F(a),x-a\right) =\left\Vert\nabla F(a)\right\Vert \left\Vert x-a\right\Vert \cos\theta(x)$$

$B$K$h$jDj5A$G$-$k!#$3$N$H$-(B

$$\lim_{x\in L_c\atop x\to a}\cos\theta(x)=0$$

$B$G$"$k$+$i(B

$$\angle R \lim_{x\in L_c\atop x\to a}\theta(x)=\frac{\pi}{2}.$$

($BCm0U(B: $x\to a$ $B$N$H$-(B $\dfrac{x-a}{\left\Vert x-a\right\Vert}$ $B<+?H$,(B $B2?$+$"$k%Y%/%H%k$K<}B+$9$k$H$$$&$o$1$G$O$J$/!"(B $B<}B+$,J]>Z=PMh$k$N$O3QEY$@$1$G$"$k!#(B)

$B0J>e$N9M;!$rGX7J$K!"

$B8=;~E@$G(B ($\angle R$) $B$O!":,5r$K$"$$$^$$$J$H$3$m$,$"$k$N$G!"(B $B!V(B$\nabla F$ $B$,K!@~%Y%/%H%k!W$H$$$&$N$O(B$BDj5A$7$?$3$H(B$B$G$"$k!#(B $B$3$NE@!"2?$+$:$k$$$H46$8$i$l$k$+$bCN$l$J$$$,(B²$B!"(B $B8e$G1"4X?tDjM}$r3X$V$H!"(B($\angle R$) $B$,:,5r$r;}$C$?e$7$FMh$k!#(B