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I recently started to go a little further in depth into studying sheaves.

I have two questions. Firstly, is the presheaf of exact forms on a smooth manifold necessarily a sheaf? I was under the impression that it wasn't. Additionally, is the space of once differentiable (but not necessarily continuously differentiable) functions together with their domains on a smooth manifold (with restriction maps being restrictions of domains) a sheaf? I'm having trouble looking at these two.

Additionally, can anyone point me to any good internet resources for learning more about using sheaves? I've got a couple of sources from a quick google search, but I'm intending to go a little further in depth this afternoon.

Thank you.