diff --git a/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf
index ebf62be..c568ada 100644
Binary files a/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf and b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf differ
diff --git a/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex
index 2dbb487..31fd1b5 100644
--- a/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex
+++ b/advanced-math/exercise/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex
@@ -120,6 +120,8 @@ $z_{xx}''=y^2f(xy)+y^2f(xy)=2y^2f(xy)$，同理根据变量对称性$z_{yy}''=2x
 
 \textbf{例题：}设$z=f(x,y)$满足$\dfrac{\partial^2z}{\partial x\partial y}=x+y$，且$f(x,0)=x$，$f(0,y)=y^2$，求$f(x,y)$。
 
+解：
+
 \section{多元函数微分应用}
 
 \subsection{空间曲线的切线与法平面}