Water Heater

If locking up & going out for the day I personally would turn of the heater & the outside mains water tap, whilst you are on site the choice is yours. 

Rate of loss of heat from a hot body to the ambient environment will always be greater if the differential between the two temperatures is greater.  As your water tank cools, the rate of loss of heat decreases.  If you maintain it at maximum temperature constantly (ie thermostat cutting in-and-out to maintain the temperature) more heat will be wasted to the environment.

if you want to discuss Newton's law of cooling we can! (which is what you're saying)  but you can't just use that in isolation and you have to look at the heat loss and how it happens, and so you're only quoting half the story. You are quoting the classic  how to keep your cup of coffee hot by adding or not adding cold milk. 

Your 'statement' is correct but your tank will be insulated so that the heat loss from it is kept to a minimum so that the heat loss you talk about doesn't apply.

Imagine a bottle and a thermos flask. You put the same amount of liquid at the same temperature in both. Both the bottle and flask have the same outside and inside temperatures. The thermos inside at at 80C and outside it could be 20C, and the same for the bottle. Now according to your statement both should lose the same amount of heat and both should cool down at the same rate? Do they? Of course they don't, the same thing applies to you caravan tank. 

Interesting that the consensus is to leave it on, I am glad I am not being wasteful with daughters electricity!  

Does the same apply to heating it by gas when off EHU?  

"Now according to your statement both should lose the same amount of heat and both should cool down at the same rate?"

No, because the 'bodies' in this case are the bottle and the thermos flask. The external surface temperature of the thermos flask will be lower and therefore the rate of heat loss will be lower. (lets assume for simplicity that the external surface area of each container is equal). The heat loss from the thermos is still however finite.

(Of course if the insulation of the hot water tank were perfect, then it would never loose any heat, and it doesn't matter whether you switch the immersion off or not   ....... except that you might end up with a tankfull of hot water which goes to waste )

I always turn mine off when going out for the day. The tank is reasonably insulated & I have added a polystyrene box cover. I find that the water after being out all day is still reasonably hot.

Sometimes I leave it on. Sometimes I switch it off. 

When we tested it with a meter there was little difference in the power used. 

Why worry........you are on holiday..............!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

The external surface temperature of the thermos flask will be lower and therefore the rate of heat loss will be lower.

Yes, well done you've got there, because it (the flask) is insulated, in the same way as your hot water tank is! but not as efficiently  

I have no idea which method is the most energy efficient (but I strongly suspect that turning off the water heater would be more efficient) but we leave it on and let the thermostat do its job - far more convenient.