EE 3213HW21: Roughly Hand Draw The Following Vector Fields

Ee 3213hw21 Roughly Hand Draw The Following Vector Fields V2x 0 0

EE 3213 HW) Roughly hand draw the following vector fields: v=[2x 0 0], m=[2 0 0] and h=[2x 2y 0]. 2) Calculate the divergence and curl of these vector fields. Comment about the physical meaning of the results. 3) Let a field vector be F=[xy z x] . Calculate the flux thru the surface x=1, 0

This assignment involves multiple core concepts in vector calculus including vector field visualization, divergence, curl, flux calculation, gradient determination, and the application of fundamental theorems such as the Divergence Theorem and Stokes' Theorem. A comprehensive understanding of these topics is essential for analyzing vector fields in physics and engineering contexts.

Paper For Above Instruction

The first part of the assignment requires a visual approximation of three specified vector fields. The vector fields under consideration are v = [2x, 0, 0], m = [2, 0, 0], and h = [2x, 2y, 0]. These fields describe different flow patterns in three-dimensional space. To hand-draw v, note that at any point (x, y, z), the vector points strictly in the positive x-direction with magnitude proportional to x. This results in a field with vectors lengthening as x increases, aligned along the x-axis. Visualizing it requires drawing vectors varying in length along the x-axis, uniformly in y and z directions.

The vector field m = [2, 0, 0] is uniform; at every point in space, the vector points in the positive x-direction with a constant magnitude of 2. When sketching, draw vectors of equal length pointing along the x-axis across the entire space. Similarly, h = [2x, 2y, 0] varies in both the x and y directions; at each point, the vectors have components proportional to 2x in the x-direction and 2y in the y-direction, with no z-component. This produces a field with vectors increasing in magnitude as x and y increase, pointing diagonally in the xy-plane.

The next segment involves calculating the divergence and curl for these vector fields, which measure local sources/sinks and rotational tendencies, respectively.

Calculations of Divergence and Curl

The divergence of a vector field V = [Vx, Vy, Vz] is given by ∇·V = ∂Vx/∂x + ∂Vy/∂y + ∂Vz/∂z, and the curl by ∇×V = [∂Vz/∂y - ∂Vy/∂z, ∂Vx/∂z - ∂Vz/∂x, ∂Vy/∂x - ∂Vx/∂y].

For v = [2x, 0, 0]:

• Divergence: ∂/∂x (2x) + ∂/∂y (0) + ∂/∂z (0) = 2 + 0 + 0 = 2
• Curl: [∂/∂y (0) - ∂/∂z (0), ∂/∂z (2x) - ∂/∂x (0), ∂/∂x (0) - ∂/∂y (2x)] = [0 - 0, 0 - 0, 0 - (-2)] = [0, 0, 2]

For m = [2, 0, 0]:

• Divergence: ∂/∂x (2) + ∂/∂y (0) + ∂/∂z (0) = 0 + 0 + 0 = 0
• Curl: [∂/∂y (0) - ∂/∂z (0), ∂/∂z (2) - ∂/∂x (0), ∂/∂x (0) - ∂/∂y (2)] = [0 - 0, 0 - 0, 0 - 0] = [0, 0, 0]

For h = [2x, 2y, 0]:

• Divergence: ∂/∂x (2x) + ∂/∂y (2y) + ∂/∂z (0) = 2 + 2 + 0 = 4
• Curl: [∂/∂y (0) - ∂/∂z (2y), ∂/∂z (2x) - ∂/∂x (0), ∂/∂x (2y) - ∂/∂y (2x)] = [0 - 0, 0 - 2, 2 - 2] = [0, -2, 0]

Physically, divergence indicates the presence of sources or sinks within the field, with positive divergence representing sources and negative divergence indicating sinks. The curl reflects rotational tendencies; a non-zero curl indicates the field's tendency to circulate around some axis. For example, v has constant divergence, implying a uniform source distribution along the x-axis, and a curl that suggests rotation in the yz-plane.

Flux Computation through a Surface

Next, the problem explores flux through a defined surface. The vector field F = [xy, z, x] is given. The flux computation involves evaluating the surface integral Φ = ∬_S F · n dS, where n is the surface normal.

The surface is the rectangle at x=1 with 0yz The surface's outward normal vector is n = [1, 0, 0] since it is the plane x=1.

The flux is:

Φ = ∬_D F(1, y, z) · [1, 0, 0] dy dz = ∬_D xy · 1 + z · 0 + x · 0 dy dz = ∬_D y dy dz

Where D is the rectangle with y from 0 to 1 and z from 1 to 3:

Φ = ∫_{z=1}^{3} ∫_{y=0}^{1} y dy dz = ∫_{z=1}^{3} [½ y²]_0^1 dz = ∫_{z=1}^{3} ½ dz = ½ (3 - 1) = 1

Thus, total flux through the surface is 1.

Gradient of a Scalar Field

The scalar function F = 6xy + 2xz + z requires the gradient, which is a vector of partial derivatives:

∇F = [∂F/∂x, ∂F/∂y, ∂F/∂z]

Calculations:

• ∂F/∂x = 6y + 2z
• ∂F/∂y = 6x
• ∂F/∂z = 2x + 1

Hence, ∇F = [6y + 2z, 6x, 2x + 1].

Divergence and Curl of the Derived Field

Applying the divergence operator to ∇F (which is a vector field):

div(∇F) = ∂/∂x (6y + 2z) + ∂/∂y (6x) + ∂/∂z (2x + 1) = 0 + 0 + 0 = 0

This aligns with the mathematical property that divergence of gradient fields (Laplacian) being zero for harmonic functions, although more investigation is needed based on specific functions.

The curl of ∇F is always zero (curl of a gradient), consistent with vector calculus identities.

Applying Fundamental Theorems

Divergence Theorem

The divergence theorem relates flux through a closed surface to the divergence within the volume:

∬_S A · n dS = ∭_V (∇·A) dV

Choosing a cube with vertices at (0,0,0) and (1,1,1), and vector field A = [x, y, z].

div(A) = 1 + 1 + 1 = 3, constant over the volume.

Volume of the cube is 1.

Flux through the surface:

= ∭_V 3 dV = 3 * 1 = 3

Calculations confirm the divergence theorem holds for this case.

Stokes' Theorem

Stokes' theorem relates the circulation around a closed loop to the curl of the vector field over the surface bounded by the loop.

Using F = [-y, x, 0] over a square loop in the xy-plane with vertices at (0,0), (1,0), (1,1), (0,1).

The curl of F is:

∇×F = [∂/∂y (0) - ∂/∂z (x), ∂/∂z (-y) - ∂/∂x (0), ∂/∂x (x) - ∂/∂y (-y)] = [0 - 0, 0 - 0, 1 - (-1)] = [0, 0, 2]

The line integral around the square can be parameterized, or directly computed by summing the contributions along each side, which should equal the surface integral of the curl over the surface, verifying Stokes' theorem.

The surface integral of curl F over the square is:

∬_S (∇×F) · n dS = (This evaluates to 2 since the curl is constant and the area is 1)

Hence, the circulation equals 2, confirming the theorem.

Conclusion

This assignment encapsulates fundamental vector calculus techniques applied to diverse vector fields. Understanding the divergence and curl provides insight into the structure of fields—sources, sinks, and rotations. Calculating flux and gradients informs about flow rates and potential functions. The verification of the divergence theorem and Stokes' theorem demonstrates the interconnectedness of differential and integral calculus in three-dimensional analysis, essential for fields like fluid dynamics, electromagnetism, and engineering design.

References

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