MAT2615 Assignment 2 solutions 2026

MAT2615 Assignment 2 solutions 2026
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CALCULUS IN HIGHER DIMENSIONS
MAT2615
Year module
Department of Mathematical Sciences
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, QUESTION 1

Given:
𝑓(𝑥, 𝑦) = 1 − 𝑥 2 − 𝑦 2



(a) Equation of contour curve 𝑪

A contour curve is defined by:

𝑓(𝑥, 𝑦) = constant


Level is:

𝑓(1,1) = 1 − 12 − 12 = 1 − 1 − 1 = −1


So:

1 − 𝑥 2 − 𝑦 2 = −1


Rearrange:

𝑥2 + 𝑦2 = 2
Answer:

𝐶: 𝑥 2 + 𝑦 2 = 2




(b) Vector perpendicular to 𝑪

Perpendicular vector = gradient:

∇𝑓 = (−2𝑥, −2𝑦)


At (1,1):

∇𝑓(1,1) = (−2, −2)


Answer:

(−2, −2)