Course Code: EMA51002
Course Name : Analytical Mathematics
Module 1: Vector Calculus
PART A CO BTL
1. Find and if 2xz 4 x 2 y at (2, -2,1). CO1 2
2. Find the unit normal vector to the surface x 2 y 2 z at (1, -2,5). CO1 2
3. CO1 1
Define solenoidal vector function. If V ( x 3 y )i ( y 2 z ) j ( x 2 z )k is
solenoidal, then find the value of .
4. Show that a vector field F ( x 2 y 2 x )i (2 xy y ) j is irrotational. CO1 2
5. For what values of ‘a’, ‘b’ and ‘c’, the vector field CO1 2
F ( x 2 y az )i (bx 3 y z ) j (4 x cy 2 z )k is irrotational.
6. Prove that div r =3. CO1 2
Evaluate (log r ) .
7. 2 CO1 2
8. Prove that curl r = 0 . CO1 2
9. State Green’s theorem. CO1 1
10. CO1 2
Using Gauss divergence theorem, prove that r . n ds 3V , where V is the volume
S
enclosed by the surface S.
PART B CO BTL
1. Find ‘a’ and ‘b’ so that the surfaces ax3 by 2z (a 3)x2 and 4x2y z 3 11 cut CO1 3
orthogonally at 2, 1, 3 .
2. Find the directional derivative of x2 y2 2z 2 at P (1, 0, 2) in the direction of CO1 3
the line PQ where Q is the point (2, 3, 4).
3. CO1 3
If F x 3i y 3 j z 3k , find div(curl F ).
4. Find the angle between the surfaces x log z y 2 – 1 and x 2 y 2 – z at the point CO1 3
(1, 1, 1).
5. If ( y 2 2 xyz 3 )i (3 2 xy x 2 z 3 ) j (6z 3 3 x 2 yz 2 )k , find . CO1 3
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Course Name : Analytical Mathematics
Module 1: Vector Calculus
PART A CO BTL
1. Find and if 2xz 4 x 2 y at (2, -2,1). CO1 2
2. Find the unit normal vector to the surface x 2 y 2 z at (1, -2,5). CO1 2
3. CO1 1
Define solenoidal vector function. If V ( x 3 y )i ( y 2 z ) j ( x 2 z )k is
solenoidal, then find the value of .
4. Show that a vector field F ( x 2 y 2 x )i (2 xy y ) j is irrotational. CO1 2
5. For what values of ‘a’, ‘b’ and ‘c’, the vector field CO1 2
F ( x 2 y az )i (bx 3 y z ) j (4 x cy 2 z )k is irrotational.
6. Prove that div r =3. CO1 2
Evaluate (log r ) .
7. 2 CO1 2
8. Prove that curl r = 0 . CO1 2
9. State Green’s theorem. CO1 1
10. CO1 2
Using Gauss divergence theorem, prove that r . n ds 3V , where V is the volume
S
enclosed by the surface S.
PART B CO BTL
1. Find ‘a’ and ‘b’ so that the surfaces ax3 by 2z (a 3)x2 and 4x2y z 3 11 cut CO1 3
orthogonally at 2, 1, 3 .
2. Find the directional derivative of x2 y2 2z 2 at P (1, 0, 2) in the direction of CO1 3
the line PQ where Q is the point (2, 3, 4).
3. CO1 3
If F x 3i y 3 j z 3k , find div(curl F ).
4. Find the angle between the surfaces x log z y 2 – 1 and x 2 y 2 – z at the point CO1 3
(1, 1, 1).
5. If ( y 2 2 xyz 3 )i (3 2 xy x 2 z 3 ) j (6z 3 3 x 2 yz 2 )k , find . CO1 3
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