As a PhD student, should I give a talk at a conference outside my supervisor's field? I am a PhD student in mathematics. Talking to some colleagues on my department I wrote a paper whose subject lies somewhat beyond the interests of my advisor. I gave a talk on it on a couple of seminars. 

Unexpectedly I was invited to give a talk on a conference in another country. I have a feeing that my advisor might be unhappy if I go there, in particular he refused to give me any funds for travel. (It might be he does not have any funds, but I still have a feeling that he does not want me to be involved in activities in other fields.) Nevertheless I am interested to present my results to a broader audience and one of the colleagues on the department I talked to agrees to help me with the travel funds.


  Would it be ok if I will go to the conference? Is it ethical with respect to my supervisor?
As @eykanal stated, the answer is simply to talk to your supervisor.

To specifically answer your questions:

1- Yes, it is more than OK to go to the conference. Being invited to an international conference is a great achievement that you should be proud of.

2- I don't see how it could be unethical to your supervisor.

I'm not sure from your question if your advisor has already refused the funds. I would suggest to not overthink this. Budget are generally tight. If your supervisor is a normal human being (most are), he's probably proud of your achievement. If for some reasons he doesn't approve, but does not clearly forbid you to go...well that's a strange situation but you can't guess his mind, can't you? If your relation is good, discussing is the only way to know.

Just remember that the goal of the PhD is to create an independent researcher out of you. Not a duplicate of your supervisor.