How can I prove the linearity of $F_*$? What does $F_*$ eat?


  If $N$ is smooth manifolds and $F: M \to N$ is a smooth map, for each $p \in M$ we define a map $F_*: T_pM \to T_{F(p)}N$, called the push-forward associated with $F$, by
  $$(F_*X)(f) = X(f \circ F).$$



Question: apache error and access logs are missing, what could be the cause and how can I recreate them?
Answer: When path references in an Apache configuration are not absolute (i.e. don't start with /) they are relative to the directory set in the ServerRoot directive. A common default for the ServerRoot is  /etc/httpd making /etc/httpd/logs your log directory. 

Most packaged Apache versions create a symbolic link from /etc/httpd/logs to /var/log/httpd. Either the actual directory in /etc/httpd is missing (or the root filesystem is full) or the symbolic link may be missing in your setup.

Apache gets started by root and opens the log files as root before dropping privileges so those directory permissions on /var/log/httpd seem correct.