The mandibular part of the maxillary artery passes down to the mandible and goes deep to its neck. This portion is located posterior the lateral pterygoid muscle.

The maxillary artery gives off five branches from its mandibular part:

Deep auricular artery - supplies the external acoustic meatus, tympanic membrane, and temporomandibular joint; goes through the squamotympanic fissure;
Anterior tympanic artery - branch to the mucous membrane of the tympanic cavity; travels through the petrotympanic fissure;
Middle meningeal artery - goes between the tensor veli palatini and lateral pterygoid; this artery reaches the middle cranial fossa via the foramen spinosum to supply its dura mater; middle meningeal artery gives a small branch to the mucous membrane of the tympanic cavity;
Inferior alveolar artery - enters the mandibular canal via the mandibular foramen and exits the canal as the mental branch via the mental foramen of the mandible; the mental branch supplies the muscles of the facial expression and the skin in the chin area; before the artery enters the mandibular canal it also gives mylohyoid branch that provides arterial blood supply to the mylohyoid muscle and anterior belly of the digastric muscle; while passing through the mandibular canal, inferior alveolar artery gives dental branches to the teeth and gingival branches to the gingiva;
Accessory meningeal artery - a small branch that provides arterial blood supply to the trigeminal ganglion, as well as the dura mater of the middle cranial fossa; goes through the foramen ovale.

Overall, all these branches enter and supply the bones of the skull.

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Mandibular part of maxillary artery