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--- math121b:02-07 [2020/02/07 00:02]
pzhou
+++ math121b:02-07 [2020/02/07 00:04]
pzhou
@@ Line 37: / Line 37: @@
 $$ I = \int \sum_i \d_{x_i} f \cdot \d_{x_i} \varphi dx_1 \cdots d x_n = \int (- \sum_i \d_{x_i}^2 f) \cdot \varphi dx_1 \cdots d x_n $$
 where we used integration by part. Thus, we get
-$$ \Delta f(x) = \sum_i\d_{x_i} ^2 f(x) $$
+$$ \Delta f(x) = \sum_i\d_{x_i}^2 f(x) $$
+So this definition makes sense.
 ** (2) If we use curvilinear cooridnate **, then we claim that
-$$ d Vol_g(u) = \sqrt{|g|} du_1\cdots du_n $$
+$$ d Vol_g(u) = \sqrt{|g|} du_1\cdots du_n \tag{1}$$
 and that
-$$ (df, d\varphi)_g = \sum_{i,j} g^{ij} \d_{u_i} f \d_{u_j} \varphi $$.
+$$ (df, d\varphi)_g = \sum_{i,j} g^{ij} \d_{u_i} f \d_{u_j} \varphi \tag{2}$$.
 Given these two claim, we get

Lecture Notes