Please help with the following solution. This is Vector Analysis (Vector Calculus). The second attachment is how the book defines divergence. 3. Find the divergence of\[\mathbf{F} = \frac{u_1}{2u_2u_3}\mathbf{e_1} + \frac{u_2}{u_3}\mathbf{e_2} + \frac{u_3}{u_1u_2}\mathbf{e_3}\]if\[h_1 = u_1, \quad h_2 = u_1u_2, \quad h_3 = u_3.\] **Mathematical Formulas for Divergence, Curl, and Laplacian**### DivergenceThe divergence of a vector field $ \mathbf{F} $ is given by:\[\nabla \cdot \mathbf{F} = \frac{1}{h_1 h_2 h_3} \left[ \frac{\partial}{\partial u_1}(F_1 h_2 h_3) + \frac{\partial}{\partial u_2}(F_2 h_1 h_3) + \frac{\partial}{\partial u_3}(F_3 h_1 h_2) \right]\]### CurlThe curl of a vector field $ \mathbf{F} $ is represented as:\[\nabla \times \mathbf{F} = \frac{1}{h_1 h_2 h_3}\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\h_1 e_1 & h_2 e_2 & h_3 e_3 \\\frac{\partial}{\partial u_1} & \frac{\partial}{\partial u_2} & \frac{\partial}{\partial u_3} \\F_1 h_1 & F_2 h_2 & F_3 h_3\end{vmatrix}\]### LaplacianThe Laplacian of a scalar function $ f $ is expressed as:\[\nabla^2 f = \frac{1}{h_1 h_2 h_3} \left[\frac{\partial}{\partial u_1}\left(\frac{h_2 h_3}{h_1} \frac{\partial f}{\partial u_1}\right) + \frac{\partial}{\partial u_2}\left(\frac{h_3 h_1}{h_2} \frac{\partial f}{\partial u_2}\right) +\frac{\partial}{\partial u_3}\left(\frac{h_1 h_2}{h_3} \frac{\partial f}{\partial u_3}\right)\right]\]---### Cylindrical Coordinates- **Displacement Vector**: \( d\mathbf{R} = dp \, e

Given the field , F=u12u2u3e1+u2u3e2+u3u1u2e3 and h1=u1, h2=u1u2, h3=u3 Divergence of F is given by,…

Answered: 3. Find the divergence of U3 -e3 U1U2…

Math

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3. Find the divergence of U3 -e3 U1U2 U1 U2 -ēj + e2 + F 2uzu3 U3 if hi = U1U2, h3 = Uz. = U1, h2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please help with the following solution. This is Vector Analysis (Vector Calculus). The second attachment is how the book defines divergence.

3. Find the divergence of

\[
\mathbf{F} = \frac{u_1}{2u_2u_3}\mathbf{e_1} + \frac{u_2}{u_3}\mathbf{e_2} + \frac{u_3}{u_1u_2}\mathbf{e_3}
\]

if

\[
h_1 = u_1, \quad h_2 = u_1u_2, \quad h_3 = u_3.
\]

**Mathematical Formulas for Divergence, Curl, and Laplacian**

### Divergence
The divergence of a vector field $ \mathbf{F} $ is given by:

\[
\nabla \cdot \mathbf{F} = \frac{1}{h_1 h_2 h_3} \left[ \frac{\partial}{\partial u_1}(F_1 h_2 h_3) + \frac{\partial}{\partial u_2}(F_2 h_1 h_3) + \frac{\partial}{\partial u_3}(F_3 h_1 h_2) \right]
\]

### Curl
The curl of a vector field $ \mathbf{F} $ is represented as:

\[
\nabla \times \mathbf{F} = \frac{1}{h_1 h_2 h_3}
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
h_1 e_1 & h_2 e_2 & h_3 e_3 \\
\frac{\partial}{\partial u_1} & \frac{\partial}{\partial u_2} & \frac{\partial}{\partial u_3} \\
F_1 h_1 & F_2 h_2 & F_3 h_3
\end{vmatrix}
\]

### Laplacian
The Laplacian of a scalar function $ f $ is expressed as:

\[
\nabla^2 f = \frac{1}{h_1 h_2 h_3} \left[
\frac{\partial}{\partial u_1}\left(\frac{h_2 h_3}{h_1} \frac{\partial f}{\partial u_1}\right) +
\frac{\partial}{\partial u_2}\left(\frac{h_3 h_1}{h_2} \frac{\partial f}{\partial u_2}\right) +
\frac{\partial}{\partial u_3}\left(\frac{h_1 h_2}{h_3} \frac{\partial f}{\partial u_3}\right)
\right]
\]

---

### Cylindrical Coordinates
- **Displacement Vector**: \( d\mathbf{R} = dp \, e

Calculus that deals with differentiation and integration of the vector field in three-dimensional Euclidean space. It deals with quantities that have both magnitude and direction.

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