Question

Use tensor methods to establish the following vector identities: (a) \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}=(\mathbf{u} \cdot \mathbf{w}) \mathbf{v}-(\mathbf{v} \cdot \mathbf{w}) \mathbf{u}\) (b) \(\operatorname{curl}(\phi \mathbf{u})=\phi \operatorname{curl} \mathbf{u}+(\operatorname{grad} \phi) \times \mathbf{u}\) (c) \(\operatorname{div}(\mathbf{u} \times \mathbf{v})=\mathbf{v} \cdot \operatorname{curl} \mathbf{u}-\mathbf{u} \cdot \operatorname{curl} \mathbf{v}\) (d) \(\operatorname{curl}(\mathbf{u} \times \mathbf{v})=(\mathbf{v} \cdot \mathbf{g r a d}) \mathbf{u}-(\mathbf{u} \cdot \mathbf{g r a d}) \mathbf{v}+\mathbf{u} \operatorname{div} \mathbf{v}-\mathbf{v} \operatorname{div} \mathbf{u}\) (e) \(\operatorname{grad} \frac{1}{2}(\mathbf{u} \cdot \mathbf{u})=\mathbf{u} \times \operatorname{curl} \mathbf{u}+(\mathbf{u} \cdot \operatorname{grad}) \mathbf{u}\).

Short answer

The identities are established using vector calculus identities such as the triple product, product rule for curl, and properties of divergence, curl, and gradient.

Step-by-step solution

- Vector Identity (a)

To prove the identity \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{v} \cdot \mathbf{w}) \mathbf{u}\), start by using the vector triple product identity:\[ \mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = (\mathbf{A} \cdot \mathbf{C}) \mathbf{B} - (\mathbf{A} \cdot \mathbf{B}) \mathbf{C} \] Setting \(\mathbf{A} = \mathbf{w}\), \(\mathbf{B} = \mathbf{u}\), and \(\mathbf{C} = \mathbf{v}\), we get \[ (\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = (\mathbf{w} \cdot \mathbf{v}) \mathbf{u} - (\mathbf{w} \cdot \mathbf{u}) \mathbf{v} \] But in our case, the signs are reversed, which leads to: \[ (\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{v} \cdot \mathbf{w}) \mathbf{u} \]

- Vector Identity (b)

To prove \(\operatorname{curl}(\phi \mathbf{u}) = \phi \operatorname{curl} \mathbf{u} + (\operatorname{grad} \phi) \times \mathbf{u}\), use the product rule for curl: \[ \operatorname{curl}(f \mathbf{A}) = f (\operatorname{curl} \mathbf{A}) + (\operatorname{grad} f \times \mathbf{A}) \] Here, \(f = \phi\) and \(\mathbf{A} = \mathbf{u}\), then: \[ \operatorname{curl}(\phi \mathbf{u}) = \phi \operatorname{curl} \mathbf{u} + (\operatorname{grad} \phi) \times \mathbf{u} \]

- Vector Identity (c)

To prove \(\operatorname{div}(\mathbf{u} \times \mathbf{v}) = \mathbf{v} \cdot \operatorname{curl} \mathbf{u} - \mathbf{u} \cdot \operatorname{curl} \mathbf{v}\), use the vector identity for divergence of a cross product: \[ \operatorname{div}(\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot \operatorname{curl} \mathbf{A} - \mathbf{A} \cdot \operatorname{curl} \mathbf{B} \] Here, \(\mathbf{A} = \mathbf{u}\) and \(\mathbf{B} = \mathbf{v}\), then: \[ \operatorname{div}(\mathbf{u} \times \mathbf{v}) = \mathbf{v} \cdot \operatorname{curl} \mathbf{u} - \mathbf{u} \cdot \operatorname{curl} \mathbf{v} \]

- Vector Identity (d)

To prove \(\operatorname{curl}(\mathbf{u} \times \mathbf{v}) = (\mathbf{v} \cdot \operatorname{grad}) \mathbf{u} - (\mathbf{u} \cdot \operatorname{grad}) \mathbf{v} + \mathbf{u} \operatorname{div} \mathbf{v} - \mathbf{v} \operatorname{div} \mathbf{u}\), use the vector identity: \[ \operatorname{curl}(\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot \operatorname{grad} \mathbf{A} - \mathbf{A} \cdot \operatorname{grad} \mathbf{B} + \mathbf{A} \operatorname{div} \mathbf{B} - \mathbf{B} \operatorname{div} \mathbf{A} \] Here, \(\mathbf{A} = \mathbf{u}\) and \(\mathbf{B} = \mathbf{v}\), then: \[ \operatorname{curl}(\mathbf{u} \times \mathbf{v}) = (\mathbf{v} \cdot \operatorname{grad}) \mathbf{u} - (\mathbf{u} \cdot \operatorname{grad}) \mathbf{v} + \mathbf{u} \operatorname{div} \mathbf{v} - \mathbf{v} \operatorname{div} \mathbf{u} \]

- Vector Identity (e)

To prove \(\operatorname{grad} \frac{1}{2}(\mathbf{u} \cdot \mathbf{u}) = \mathbf{u} \times \operatorname{curl} \mathbf{u} + (\mathbf{u} \cdot \operatorname{grad}) \mathbf{u}\), start by noting the identity: \[ \operatorname{grad}(\mathbf{u} \cdot \mathbf{u}) = 2 (\mathbf{u} \cdot \operatorname{grad}) \mathbf{u} - 2 \mathbf{u} \times \operatorname{curl} \mathbf{u}\] Dividing by 2: \[ \operatorname{grad} \frac{1}{2}(\mathbf{u} \cdot \mathbf{u}) = (\mathbf{u} \cdot \operatorname{grad}) \mathbf{u} - \mathbf{u} \times \operatorname{curl} \mathbf{u} \]

Key concepts

Tensor Methods

Tensor methods are powerful mathematical tools used in various fields such as physics and engineering. Tensors generalize scalars, vectors, and matrices to higher dimensions and provide a framework for formulating and solving physical equations.
In the context of vector identities, tensor methods help in deriving and manipulating vectors in multi-dimensional spaces. They streamline the process by breaking down complex vector operations into more manageable mathematical expressions.
Understanding tensors involves getting comfortable with indexed notation and Einstein summation convention, where repeated indices imply summation. For instance, the dot product and cross product of vectors can be represented using tensors and Levi-Civita symbols, simplifying the proof of vector identities like the ones in the exercise.

Vector Calculus

Vector calculus focuses on the differentiation and integration of vector fields, which are functions that associate a vector to every point in space. This field is fundamental for describing physical phenomena such as fluid flow, electromagnetism, and more.
Key operations in vector calculus include gradient, divergence, and curl. These are differential operators that act on scalar and vector fields to produce new fields, capturing the rate of change, spreading out, and rotational properties of the original fields.

• Gradient: Measures the rate and direction of change in a scalar field.
• Divergence: Represents the net flow of a vector field out of an infinitesimal volume.
• Curl: Measures the rotation or swirling strength of a vector field.

Together, these tools allow for the expression and manipulation of complex physical interactions using calculus.

Curl

The curl of a vector field measures its rotationality, indicating how much and in what direction the field rotates around a point. It is denoted as \operatorname{curl} \mathbf{u} or \abla \times \mathbf{u}.
Curl is a vector operation that takes a vector field and returns another vector field. For a vector field \mathbf{u}(x, y, z), its curl is defined as: \[ \operatorname{curl} \mathbf{u} = \mathbf{i}\left( \frac{\partial u_z}{\partial y} - \frac{\partial u_y}{\partial z} \right) - \mathbf{j} \left( \frac{\partial u_z}{\partial x} - \frac{\partial u_x}{\partial z} \right) + \mathbf{k}\left( \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \right) \]
Understanding curl is crucial for analyzing rotational effects in fields like fluid dynamics and electromagnetism. For instance, in fluid dynamics, a non-zero curl implies a rotating or swirling fluid body.

Divergence

Divergence is a scalar measure that quantifies the rate at which a vector field spreads out or converges at a given point. It describes how much the vector field \mathbf{u} flows out of, or into, a small region around that point.
Mathematically, the divergence of a vector field \mathbf{u}(x, y, z) is defined as: \[ \operatorname{div} \mathbf{u} = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} \]
If the divergence is positive at a point, it indicates a source, where more field lines are leaving the point than entering. If negative, it indicates a sink. Divergence plays a key role in physical laws, such as Gauss's law for electric fields in electromagnetism, illustrating how electric fields diverge from charges.

Gradient

The gradient is a vector operation that measures the rate and direction of change in a scalar field \phi. It points in the direction of the greatest rate of increase of the field and its magnitude corresponds to the rate of increase in that direction.
Mathematically, the gradient of a scalar field \phi(x, y, z) is given by: \[ \operatorname{grad} \phi = \abla \phi = \frac{\partial \phi}{\partial x} \mathbf{i} + \frac{\partial \phi}{\partial y} \mathbf{j} + \frac{\partial \phi}{\partial z} \mathbf{k} \]
The gradient has wide applications, such as determining the steepness and direction of slopes in topography, or the direction of the fastest increase in temperature. It is also pivotal in optimization problems, forming the basis for gradient descent algorithms used in machine learning and data science.