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how do the principles of convection, conduction, and radiation explain how the water in the saucepan gets hot?

The radiation of the fire causes the saucepan to heat up. the conduction of the saucepan then helps to distribute the heat over the saucepan and to give heat to the water. the convection of the water helps distributing the heat within the water, till all the water is boiling

The stove gets hot and through conduction, the surface of the saucepan also becomes hot, because it is in direct contact with the stove surface.
The water in the saucepan then becomes hot through convection because the hotter fluid rises and the colder, denser water sinks and becomes hot as well.