Consider a system of the form r = F(r), r QC R. Recall that a function...

 

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Consider a system of the form r = F(r), r QC R. Recall that a function V: QR is called potential energy, if it fulfils VV=-F. In this exercise you calculate some common "F's" as given by the gradient of a given potential. (a) [1 mark] Let a R and consider V: R3R, V(r) = a r. Prove that VV = a. (b) [2 marks] Let a > 0 and consider V : R R, V(r) = 1, where r = |r. Prove that a r VV (r): = (c) [2 marks] Let f : RR and consider V: R R, V(r) = f(r), where r = |r|. Prove that VV(r) = f(r) r Consider a system of the form r = F(r), r QC R. Recall that a function V: QR is called potential energy, if it fulfils VV=-F. In this exercise you calculate some common "F's" as given by the gradient of a given potential. (a) [1 mark] Let a R and consider V: R3R, V(r) = a r. Prove that VV = a. (b) [2 marks] Let a > 0 and consider V : R R, V(r) = 1, where r = |r. Prove that a r VV (r): = (c) [2 marks] Let f : RR and consider V: R R, V(r) = f(r), where r = |r|. Prove that VV(r) = f(r) r