I would like to know if there are different ways in Mandarin or Cantonese to express the causative, examplified by the help of the sentence "to make s.o. jump".
The causative action can be expressed in many ways, the most common one in everyday Mandarin Chinese is 让, which is used together with the verb that expresses the action.
Make X jump can be 让X跳 or 让X跳起来 depending on the context.
To further elaborate the causative:
In addition 使 and 叫 can also be used.
使 means to cause, to enable or to apply/use/employ, etc. and is often used in written language, e.g. 什么使你这样想呢? What makes you think this way?
叫 in itself means to call, to summon or to shout but can be used to express causative when someone was explicitly ordered to do something, e.g. 大夫叫他卧床一两天。The doctor ordered(/instructed) him to stay in bed for a day or two.
In some cases you can mutually replace 让, 使, 叫, in some cases not. For example if you make someone jump by ordering him/her to jump it's 叫, because you instruct him/her verbally to do so. On the other hand if you are acting carelessly e.g. with a burning torch or a sharp knife and you cause the other person jump to avoid getting hurt, it cannot be 叫, because you are not instructing or calling on this person to do something.
Also, note that when you see these verbs, they not necessarily express causative, all of these characters have their own set of meanings, but can be used to express to make someone do something.