the hilly part of a city, often an upscale or quiet residential area
この町は山の手で、古い洋館が多い。
This neighborhood is on the hill (yamano-te) and has many old Western-style houses.
山の手の住宅街は静かで緑が多い。
The hillside residential area is quiet and has a lot of greenery.
彼は山の手の育ちだから、物腰が落ち着いている。
He was raised in an upper-class neighborhood, so his manner is composed.
specifically the Yamanote residential districts of Tokyo (e.g., Yotsuya, Aoyama, Akasaka, Azabu), historically upscale neighborhoods
彼は青山の山の手に家を持っている。
He owns a house in the Yamanote (hilly) area of Aoyama.
東京の山の手は昔、侍屋敷や大名屋敷が多かった。
Tokyo's Yamanote area used to have many samurai residences and feudal lord mansions.
山の手（四谷や赤坂など）は高級住宅地として知られている。
The Yamanote areas (such as Yotsuya and Akasaka) are known as upscale residential districts.
a place near the mountains; an area along the mountainside
この村は山の手に位置しているので、冬は雪が多い。
This village is located on the mountainside, so it gets a lot of snow in winter.
ハイキングコースは山の手の道を通る。
The hiking trail goes along the mountainside.