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...Prepared for submission to jhep vector calculus ia b c allanacha j m evansa adepartment of applied mathematics and theoretical physics centre mathemati cal sciences university cambridge wilberforce road cb wa united kingdom e mail allanach damtp cam ac uk abstract these are lecture notes the tripos part course there four examples sheets this pretty much complete books see schedules a list but particularly mathematical methods engineering cup by riley hobson bence analysis cartesian tensors bourne kendall nelson thomas future lecturers please feel free use modify resource if you do though append your name current author copyright not be quoted or reproduced without permission contents derivatives coordinates dierentiation using notation function scalar position gradient directional chain rule particular case coordinate dierentiable functions n general version inverse systems curves line integrals parameterised tangents arc length fields denitions comments sums gradients exact dierential...