**Title: Analyzing Vector Fields as Gradients of Scalar Fields****Introduction:**Explore which vector fields can be expressed as the gradient of a scalar field. This concept is fundamental in vector calculus and is broadly applied in fields like physics and engineering.**Given Functions:**1. **$ f(r) \hat{r} $**: - Description: $ f(r) $ is a continuous and differentiable function of $ r $. - Context: Assume working in 3D spherical coordinates.2. **$ g(\phi) \hat{\phi} $**: - Description: $ g(\phi) $ is continuous and differentiable. - Context: Assume working in 3D spherical coordinates.3. **$ xy \hat{x} + xy \hat{y} $**: - Nature: This is a simple 2D vector field.4. **$ \sin\phi \cos\phi \hat{\phi} $**: - Context: Assume working in cylindrical coordinates.5. **$-\frac{xy^2 \hat{x} + x^2 y \hat{y}}{(x^2+y^2)^{3/2}} $**: - This expression provides the vector field components in 2D Cartesian coordinates.**Objective:**Determine which of these functions can be represented as the gradient of a scalar field.

Answered: Which of the following functions can be…

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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**Title: Analyzing Vector Fields as Gradients of Scalar Fields**

**Introduction:**
Explore which vector fields can be expressed as the gradient of a scalar field. This concept is fundamental in vector calculus and is broadly applied in fields like physics and engineering.

**Given Functions:**

1. **$ f(r) \hat{r} $**:
- Description: $ f(r) $ is a continuous and differentiable function of $ r $.
- Context: Assume working in 3D spherical coordinates.

2. **$ g(\phi) \hat{\phi} $**:
- Description: $ g(\phi) $ is continuous and differentiable.
- Context: Assume working in 3D spherical coordinates.

3. **$ xy \hat{x} + xy \hat{y} $**:
- Nature: This is a simple 2D vector field.

4. **$ \sin\phi \cos\phi \hat{\phi} $**:
- Context: Assume working in cylindrical coordinates.

5. **$-\frac{xy^2 \hat{x} + x^2 y \hat{y}}{(x^2+y^2)^{3/2}} $**:
- This expression provides the vector field components in 2D Cartesian coordinates.

**Objective:**
Determine which of these functions can be represented as the gradient of a scalar field.

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A coordinate system is a convention that is used to represent a point uniquely using a set of numbers. If the coordinate system is three-dimensional, three numbers(ordered trail) are needed to uniquely represent a point. Three common coordinate systems used to represent a point in the plane are the cartesian coordinate system, spherical polar coordinate system, and cylindrical coordinate system.

A vector is a quantity that has both magnitude and direction. The general form of a vector in the cartesian coordinate system is represented as $\vec{A} = A_{1} \hat{x} + A_{2} \hat{y} + A_{3} \hat{z}$ , where $A_{1}, A_{2}, A_{3}$ are called the components of the vectors, and $\hat{x}, \hat{y}$ and $\hat{z}$ are the units vectors in the direction of x, y and z respectively. Similarly, general form of a vector in the spherical polar coordinate system is represented as $\vec{A} = A_{1} \hat{r} + A_{2} \hat{θ} + A_{3} \hat{ϕ}$ , and $\hat{θ}, \hat{ϕ}$ and $\hat{r}$ are the unit vectors in the spherical polar coordinate system.