Vector Analysis-Gauss and Stokes' Theorems, Green's Identities [IAS (Admin.) Mains Mathematics]: Questions 1 - 8 of 19

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Question 1

Appeared in Year: 2015

Describe in Detail Subjective▾

Evaluate , where is the reactangle with vertices [CS (Main) Paper 1]

Edit

Explanation

  • By Green՚s Theorem in plane,

  • Here

… (101 more equations, 3 figures) …

Question 2

Appeared in Year: 2012

Describe in Detail Subjective▾

If evaluate

Where S is the surface of the sphere above the plane. (Paper I)

Edit

Explanation

  • The boundary c of the surface s is the circle
  • The parametric equation of c are

… (27 more equations) …

Question 3

Appeared in Year: 2011

Describe in Detail Subjective▾

If , calculate the double integral over the hemisphere given by (Paper I)

Edit

Explanation

  • Here is the surface of the sphere lying above the plane. Let c be one foundry ABED of the surface S. Then the curve C is a circle in the plane and equation of C are

… (97 more equations, 3 figures) …

Question 4

Appeared in Year: 2012

Describe in Detail Subjective▾

Verify Green՚s theorem in the plane for

Where c is the closed curve of the region bounded by (Paper I)

Edit

Explanation

By Green՚s theorem in a plane

C is the Closed Curve of the Region

… (117 more equations, 8 figures) …

Question 5

Appeared in Year: 2009

Describe in Detail Subjective▾

Using divergence Theorem, evaluate

Where and is the surface of the sphere (Paper I)

Edit

Explanation

  • Let be denotes the entire surface of the sphere and let be the volume enclosed by

    i.e..

… (108 more equations) …

Question 6

Appeared in Year: 2010

Describe in Detail Subjective▾

Verify Green՚s Theorem for the path of integration being the boundary of the square whose vertices are (Paper I)

Edit

Explanation

By green՚s Theorem in plane,

Curve C is the Square
  • We have to verify this result
  • No…

… (82 more equations, 4 figures) …

Question 7

Appeared in Year: 2007

Describe in Detail Subjective▾

Determine by using stoke՚s theorem, where c is the curve defined by

That starts from the point and goes at first below the z-plane (Paper I)

Edit

Explanation

  • The curve c is given by

… (177 more equations, 6 figures) …

Question 8

Appeared in Year: 2014

Describe in Detail Subjective▾

Verify Gauss divergence theorem for the vector

Taken over the cube (Paper I)

Edit

Explanation

  • Let be the surface of the cube bounded by the planes
  • Let v be the volume enclosed by s then we have to prove

… (127 more equations, 3 figures) …