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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1993 • Question 8

Question

Math 6410 • ODE Fall 1993 • Question 8

If \(T\) is a distribution on \(G\), define \(\partial ^\alpha T\), where \(\alpha \) is a multi-index. Show that if \(f\in C^1(G)\), then \(f\) and its partial derivatives \(\partial f/\partial x_i\), \(1\le i\le n\), define unique distributions \(T\), \(T_i\), \(1\le i\le n\), on \(G\) and \(\partial ^\alpha T=T_i\); \(\alpha =(0,\ldots ,1,0,\ldots ,0)\), where the \(1\) appears in the \(i\)th place.

Raw LaTeX

If \(T\) is a distribution on \(G\), define \(\partial^\alpha T\), where
\(\alpha\) is a multi-index. Show that if \(f\in C^1(G)\), then \(f\) and its partial
derivatives \(\partial f/\partial x_i\), \(1\le i\le n\), define unique distributions
\(T\), \(T_i\), \(1\le i\le n\), on \(G\) and \(\partial^\alpha T=T_i\);
\(\alpha=(0,\ldots,1,0,\ldots,0)\), where the \(1\) appears in the \(i\)th place.

Topic Tags

Distributions and Weak Solutions