Vector Calculus NAT Level - 1 - Physics MCQ

To evaluate the line integral:  ∮ A · dr

where: A = (3x² + 6y) î - 14yz ĵ + 20xz² k̂

along the curve given by the parametric equations:

x = t, y = t², z = t³, 0 ≤ t ≤ 1

Step 1: Compute dr

The differential displacement vector is:

dr = (dx/dt) dt î + (dy/dt) dt ĵ + (dz/dt) dt k̂

We compute:  dx/dt = 1, dy/dt = 2t, dz/dt = 3t²

So, dr = (1 î + 2t ĵ + 3t² k̂) dt

Step 2: Compute A · dr

Substituting x = t, y = t², and z = t³ into A:

Aₓ = 3t² + 6t² = 3t² + 6t² = 9t²

Aᵧ = -14(t²)(t³) = -14t⁵

A_z = 20(t)(t³)² = 20t⁷

Now compute the dot product:

A · dr = (9t²)(1) + (-14t⁵)(2t) + (20t⁷)(3t²)

= 9t² - 28t⁶ + 60t⁹

Step 3: Compute the Integral

∫₀¹ (9t² - 28t⁶ + 60t⁹) dt

Computing each term separately:

∫₀¹ 9t² dt = 9 · (t³/3) |₀¹ = 9 · (1/3) = 3

∫₀¹ 28t⁶ dt = 28 · (t⁷/7) |₀¹ = 28 · (1/7) = 4

∫₀¹ 60t⁹ dt = 60 · (t¹⁰/10) |₀¹ = 60 · (1/10) = 6

Step 4: Compute the Final Result

3 - 4 + 6 = 5

Final Answer:

A · dr = 5

The correct answer is: 4.187

= –0.858
The correct answer is: -0.858

The conservative field vector 

The correct answer is: 4.141

Here is the plain text version with superscripts and subscripts:

Step 1: Compute the second derivatives
Second derivative with respect to x:
V = x² y² z²
∂V/∂x = 2x y² z²
∂²V/∂x² = 2 y² z²

Second derivative with respect to y:
∂V/∂y = 2x² y z²
∂²V/∂y² = 2 x² z²

Second derivative with respect to z:
∂V/∂z = 2x² y² z
∂²V/∂z² = 2 x² y²

Step 2: Add the second derivatives
Now, the Laplacian is the sum of these second derivatives:

∇²V = 2 y² z² + 2 x² z² + 2 x² y²

Step 3: Evaluate at the point (x, y, z) = (√2, √2, √2)
Substitute x = √2, y = √2, and z = √2 into the expression for the Laplacian:

∇²V = 2(√2)²(√2)² + 2(√2)²(v2)² + 2(v2)²(v2)²
Since (√2)² = 2, this simplifies to:

∇²V = 2 × 2 ×2 + 2 × 2 × 2 + 2 × 2 × 2 = 24

Thus, the Laplacian of the field V at the point (√2,√2,√ 2) is:

∇²V = 24

= m(m + 1)r^m-2
Hence value of  α = 2.

The correct answer is: 2

⇒ Angle between each pair is 120º

Now,  

= 54 – 10 – 10 – 25 = 9

The correct answer is: 3

By Gauss divergence theorem

The correct answer is: 226.194

The correct answer is: 6.282

Magnitude of temperature gradient

The correct answer is: 4.898