Differential Geometry

JKPSC Assistant Professor (Mathematics) · Mathematics · 48 practice questions with answers and explanations: 4 Easy, 38 Medium, 6 Hard.

Sample questions

Sample Easy question 1

The unit tangent vector of a regular curve is:

  1. α′(t) × α″(t)
  2. α″(t) ÷ |α″(t)|
  3. α(t) ÷ |α(t)|
  4. α′(t) ÷ |α′(t)|
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Answer: D. A regular curve has α′ ≠ 0.

Sample Easy question 2

The curvature of a straight line is:

  1. Zero
  2. Undefined
  3. Infinite
  4. One
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Answer: A. A line does not bend.

Sample Easy question 3

The curvature of a circle of radius r is:

  1. r²
  2. r
  3. 1 ÷ r
  4. 2πr
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Answer: C. A smaller circle bends more sharply.

Sample Medium question 4

The first fundamental form of a surface measures:

  1. Lengths and angles on the surface
  2. Only its curvature
  3. Its volume
  4. Its torsion
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Answer: A. It is the restriction of the inner product to the tangent plane.

Sample Medium question 5

The Gauss map assigns to each point of an oriented surface its:

  1. Unit normal vector
  2. Arc length
  3. Tangent plane area
  4. Principal curvature
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Answer: A. Its derivative is the shape operator.

Sample Medium question 6

The Gaussian curvature K of a surface is:

  1. The ratio of the principal curvatures
  2. The sum of the principal curvatures
  3. The product of the principal curvatures
  4. The mean of the principal curvatures
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Answer: C. The mean curvature H is their average.

Sample Hard question 7

The Gauss–Bonnet theorem relates the total Gaussian curvature of a compact surface to its:

  1. Volume
  2. Euler characteristic
  3. Area
  4. Mean curvature
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Answer: B. ∫K dA = 2π χ.

Sample Hard question 8

The torsion of the helix (a cos t, a sin t, bt) is:

  1. 1 ÷ b
  2. b ÷ a
  3. a ÷ (a² + b²)
  4. b ÷ (a² + b²)
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Answer: D. Both curvature and torsion are constant.

Sample Hard question 9

The Gaussian curvature of a sphere of radius r is:

  1. r²
  2. 0
  3. 1 ÷ r
  4. 1 ÷ r²
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Answer: D. Both principal curvatures equal 1 ÷ r.

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