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Correct option is (b) 2

For explanation: Div (F) = 4 + 7 + 1 = 12. The divergence theorem gives ∫∫∫(12).DV, where dV is the volume of the cone πr^3h/3, where r = 1/2π m and h = 4π^2 m. On substituting the RADIUS and height in the triple INTEGRAL, we get 2 units.

The correct option is (c) 39

To explain I would say: DIV (F) = 3 + 2y + x. By divergence theorem, the triple integral of Div F in the REGION is ∫∫∫ (3 + 2y + x) dx dy dz. On integrating from x = 0->1, y = 0->3 and z = 0->2, we get 39 UNITS.

Right answer is (b) 0

For EXPLANATION I WOULD say: Since the divergence of the function is zero, the TRIPLE integral LEADS to zero. The GAUSS theorem gives zero value.

Correct CHOICE is (b) 3

For EXPLANATION: The POSITION vector in Cartesian system is GIVEN by R = X i + y j + z k. Div(R) = 1 + 1 + 1 = 3. By divergence theorem, ∫∫∫3.dV, where V is a cube with x = 0->1, y = 0->1 and z = 0->1. On integrating, we get 3 units.

Right choice is (d) SURFACE to volume integral

The explanation is: The divergence theorem for a FUNCTION F is given by ∫∫ F.dS = ∫∫∫ Div (F).dV. THUS it CONVERTS surface to volume integral.

Correct answer is (b) 180π

To elaborate: We could PARAMETERISE surface and find surface INTEGRAL, but it is wise to USE divergence THEOREM to get FASTER results. The divergence theorem is given by ∫∫ F.dS = ∫∫∫ Div (F).dV

Div (3x i + 2y j) = 3 + 2 = 5. Now the volume integral will be ∫∫∫ 5.dV, where dV is the volume of the sphere 4πr^3/3 and r = 3units.Thus we get 180π.

The correct choice is (c) -2

For explanation I WOULD say: ∫∫(dG/dx – dF/DY)dx dy = ∫∫(2x – 2y)dx dy. On integrating for x = 0->1 and y = 0->2, we get Green’s value as -2.

Correct option is (a) True

To ELABORATE: The Shoelace THEOREM is used to find the area of polygon using cross MULTIPLES. This can be verified by dividing the polygon into TRIANGLES. It is a special case of Green’s theorem.

The correct answer is (b) Stoke’s theorem

The best I can explain: The Green’s theorem is a special CASE of the Kelvin- STOKES theorem, when applied to a region in the x-y PLANE. It is a widely used theorem in MATHEMATICS and physics.

Right option is (B) TWO dimensional

To ELABORATE: Since GREEN’s theorem converts line integral to surface integral, we get the value as two dimensional. In other words the functions are variable with RESPECT to x,y, which is two dimensional.

The correct ANSWER is (d) Does not EXIST

To elaborate: GREEN’s theorem is valid only for continuous functions. Since the GIVEN functions are discrete, the theorem is invalid or does not exist.

Right answer is (B) Anticlockwise

To explain: The Green’s theorem calculates the area TRAVERSED by the functions in the REGION in the anticlockwise direction. This converts the LINE integral to surface integral.

Right OPTION is (a) 0

The best I can EXPLAIN: ∫∫(dG/dx – dF/DY)dx dy = ∫∫(0 – 0)dx dy = 0. The value of GREEN’s theorem gives ZERO for the functions given.

The correct choice is (d) 5000

The explanation is: The current density is GIVEN by, J = σE. To find conductivity, σ = J/E = 1/200 X 10^-6 = 5000.

Right option is (B) 2/3

The best I can explain: From STOKE’s theorem, we can calculate P = E X I = ∫ E. J ds

= 2∫ x^2 DX as x = 0->1. We get P = 2/3 units.

The correct choice is (c) Continuous PARTIAL derivatives

The best explanation: The Green’s theorem states that if L and M are FUNCTIONS of (x,y) in an open region CONTAINING D and having continuous partial derivatives then,

∫ (F dx + G dy) = ∫∫(dG/dx – dF/dy)dx dy, with path TAKEN anticlockwise.

The correct option is (a) 0.57

The EXPLANATION: We can compute the ENERGY stored in a capacitor from Stoke’s theorem as 0.5Cv^2. Thus GIVEN energy is 0.5 X 12 X v^2. We get v = 0.57 volts.

The CORRECT CHOICE is (d) 16

To explain: From STOKE’s theorem, we can calculate energy stored in an INDUCTOR as 0.5Li^2. E = 0.5 X 2 X 4^2 = 16 units.

Right choice is (d) Curl free

Easy explanation: SINCE curl is required, we need not BOTHER about DIVERGENCE property. The curl of the FUNCTION will be i(0-0) – j(0-0) + k(0-0) = 0. The curl is zero, thus the function is said to be irrotational or curl free.

Correct option is (d) Stoke’s and Green’s theorem

The best explanation: The Stoke’s theorem is given by ∫A.dl = ∫∫ CURL (A).DS. Green’s theorem is given by, ∫ F dx + G DY = ∫∫ (dG/dx – dF/dy) dx dy. It is clear that both the theorems convert line to surface INTEGRAL.

Correct answer is (c) CURL

Easy explanation: ∫A.dl = ∫∫ Curl (A).ds is the expression for Stoke’s THEOREM. It is CLEAR that the theorem uses curl operation.

Correct option is (d) 0.275

Explanation: Del^2(V) = -ρv/εo= +106

On INTEGRATING TWICE with respect to X, V = 106. (x^2/2) + C1x + C2.

Substitute the boundary conditions, x = 0, V = 0 and x = 1mm, V = 2V in V,

C1 = 1500 and C2 = 0. At x = 0.5mm, we get, V = 0.875V.

The correct answer is (a) Yes

Easy explanation: (Del)^2V = 0

(Del)^2V = (Del)^2(25 sin θ), which is not equal to ZERO. Thus the FIELD does not satisfy Laplace equation.

Right choice is (b) False

To explain: The Poisson equation is a general case for LAPLACE equation. If volume charge DENSITY exists for a FIELD, then (Del)2V= -ρv/ε, which is CALLED Poisson equation.

Correct answer is (a) State True/False.

Best explanation: The integral FORM of Gauss LAW is ∫∫∫ ρv dv = V. THUS differential or point form will be DIV(V) = ρv.

The CORRECT option is (c) volume of CUBE

To elaborate: The volume INTEGRAL gives the volume of a vector in a REGION. Thus volume of a cube can be computed.

The correct choice is (d) 6400 π

Easy explanation: Q = D.ds = ∫∫∫ Div (D) DV, where RHS needs to be computed.

The divergence of D given is, Div(D) = 10 ρ^2 and dv = ρ dρ dφ dz. On integrating, ρ = 0->4, φ = 0->2π and Z = 0->5, we get Q = 6400 π.

The CORRECT answer is (c) 75 π

The explanation is: D.ds = ∫∫∫ Div (D) dv, where RHS needs to be computed.

The divergence of D given is, Div(D) = 5r and dv = R^2 sin θ dr dθ dφ. On integrating, r = 1->2, φ = 0->2π and θ = 0->π, we get Q = 75 π.

Correct choice is (B) 588.9

To elaborate: ∫∫ D.ds = ∫∫∫ Div (D) DV, where RHS needs to be computed.

The divergence of D GIVEN is, Div(D) = 5r and dv = R^2 sin θ dr dθ dφ. On integrating, r = 0->4, φ = 0->2π and θ = 0->π/4, we get Q = 588.9.

The correct option is (b) ρ(4πa^3/3)

To explain: The CHARGE enclosed by the sphere is Q = ∫∫∫ ρ dv.

Where, dv = R^2 sin θ dr dθ dφ and on INTEGRATING with r = 0->a, φ = 0->2π and θ = 0->π, we get Q = ρ(4πa^3/3).

Right CHOICE is (a) True

Easy EXPLANATION: Volume INTEGRAL INTEGRATES the independent quantities by THREE times. Thus it is said to be three dimensional integral or triple integral.

Right option is (a) True

The best I can explain: The triple integral, as the NAME suggests INTEGRATES the function/quantity THREE times. This GIVES volume which is the PRODUCT of three independent quantities.

The CORRECT choice is (d) CHARGE and flux

Explanation: Gauss law STATES that the ELECTRIC flux passing through any closed SURFACE is equal to the total charge enclosed by the surface. Thus, it is given by, ψ = ∫∫ D.ds= Q, where the divergence theorem computes the charge and flux, which are both the same.

Correct option is (b) 12

To explain I WOULD say: While evaluating SURFACE INTEGRAL, there has to be two variables and one constant compulsorily. ∫∫D.ds = ∫∫Dx=0 DY dz + ∫∫Dx=1 dy dz + ∫∫Dy=0 dx dz + ∫∫Dy=2 dx dz + ∫∫Dz=0 dy dx + ∫∫Dz=3 dy dx. Put D in equation, the integral value we get is 12.

Right option is (c) 5/3

The BEST explanation: A cuboid has six faces. ∫∫A.ds = ∫∫Ax=0 dy dz + ∫∫Ax=1 dy dz + ∫∫Ay=0 DX dz + ∫∫Ay=1 dx dz + ∫∫Az=0 dy dx + ∫∫Az=1 dy dx. SUBSTITUTING A and INTEGRATING we get (1/3) + 1 + (1/3) = 5/3.

The correct answer is (c) 75π

To elaborate: ∫∫ ( 5r^2/4) . (r^2 sin θ dθ dφ), which is the integral to be EVALUATED. SINCE it is double integral, we need to keep only two variables and one constant compulsorily. Evaluate it as two integrals KEEPING r = 1 for the first integral and r = 2 for the second integral, with φ = 0→2π and θ = 0→ π. The first integral value is 80π, whereas second integral GIVES -5π. On summing both integrals, we get 75π.

Right choice is (d) 6400 π

Explanation: ∫∫ D.ds = ∫∫ (10ρ^3/4).(ρ dφ dz), which is the integral to be EVALUATED. PUT ρ = 4M, z = 0→5 and φ = 0→2π, the integral EVALUATES to 6400π.

The correct choice is (c) 588.9

To explain: ∫∫ ( 5r^2/4) . (R^2 sin θ dθ dφ), which is the INTEGRAL to be evaluated.

Put r = 4m and SUBSTITUTE θ = 0→ π/4 and φ = 0→ 2π, the integral EVALUATES to 588.9.

Correct answer is (a) True

To elaborate: GAUSS law, Q = ∫∫D.ds

By considering area of a SPHERE, ds = r^2sin θ dθ dφ.

On integrating, we GET Q = 4πr^2D and D = εE, where E = F/Q.

Thus, we get Coulomb’s law F = Q1 x Q2/4∏εR^2.

Correct choice is (B) Area

To EXPLAIN: Surface integral is used to COMPUTE area, which is the product of TWO quantities length and breadth. THUS it is two dimensional integral.