aufsteigen
to rise
To move physically upward; to gain height.
Heisse Luft steigt nach oben auf.
Hot air rises upward.

to rise / mount
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to rise
To move physically upward; to gain height.
Heisse Luft steigt nach oben auf.
Hot air rises upward.

'Aufsteigen' (to rise / mount) is a separable verb. The prefix 'auf-' separates from the verb in a main clause and stands at the end of the sentence.
Der Heißluftballon steigt langsam auf.
The hot-air balloon rises slowly.
Since 'aufsteigen' (to rise / mount) describes movement or a change of state, the perfect tense is always formed with the auxiliary verb 'sein' (to be).
Der Nebel ist am Morgen aufgestiegen.
The fog rose in the morning.
When climbing onto a vehicle or mount, you use the preposition 'auf' with the accusative.
Vorsichtig stieg sie auf das unruhige Pferd auf.
Carefully, she mounted the restless horse.
To express a promotion at work, the preposition 'zu' with the dative is used to name the new position.
Nach nur zwei Jahren ist er zum Abteilungsleiter aufgestiegen.
After only two years, he rose to department head.
In a sports context, the preposition 'in' with the accusative is used to indicate the higher league that is reached.
Der Verein ist letztes Jahr in die zweite Liga aufgestiegen.
The club was promoted to the second division last year.
When feelings, thoughts, or sensations 'aufsteigen' (rise up), the subject is the feeling itself. The person who feels it is often named with the preposition 'in' and dative.
Bei seinen Worten stieg Wut in mir auf.
At his words, anger rose up in me.
to fall
'fallen' and 'aufsteigen' both describe movement in terms of height, but in opposite directions. 'fallen' means that something or someone moves downward, often without control. 'aufsteigen' means that something or someone moves upward. So 'fallen' fits a glass that goes from the table to the floor, or a person who falls. 'aufsteigen' fits smoke, a balloon, or a person who goes or climbs upward.

Über dem heißen Tee steigt langsam Dampf auf.
Steam slowly rises above the hot tea.
Beim Aufräumen ist mir ein Glas aus der Hand gefallen.
While tidying up, I dropped a glass.