Show that \(\boldsymbol{\nabla} \times(f \mathbf{A})=(\boldsymbol{\nabla} f) \times \mathbf{A}+f(\mathbf{\nabla} \times \mathbf{A})\), where \(/\) is scalar and \(\mathbf{A}\) is a vector.

In the diverse toolbox of vector calculus, certain identities provide critical shortcuts to simplifying complex expressions. A vector calculus identity typically involves operations like the dot product, cross product, gradient, divergence, or curl, applied to vectors and scalars in various combinations. The identity in question combines the curl operation with the cross product of a scalar field and a vector field.

Understanding this identity is instrumental in fields such as electromagnetism and fluid dynamics, where such combinations frequently occur. Recognizing the separate terms in the identity — the scalar field times the curl of the vector field and the gradient of the scalar field cross-multiplied with the vector field — allows students and professionals to efficiently manipulate the equations of physics and engineering.

A cornerstone of calculus, the product rule for differentiation, is a method to find the derivative of two multiplied functions. It asserts that the derivative of a product is not simply the product of the derivatives. Instead, you should take the derivative of the first function and multiply it by the second function, and then add the product of the first function and the derivative of the second function.

Mathematically expressed, for functions u(x) and v(x), the rule is: \(\frac{d}{dx}(u(x)v(x)) = u'(x)v(x) + u(x)v'(x).\) In the context of vector calculus, the product rule extends to vector fields and scalar fields, allowing for the expansion of terms like \(abla \times (f \mathbf{A})\) into expressions that reveal the nature of their interaction.

The Levi-Civita symbol, represented by \(\epsilon_{ijk}\), is a beautifully succinct way to express the permutation of indices in three-dimensional space. It is defined as 1 when (i, j, k) is an even permutation of (1, 2, 3), -1 when it is an odd permutation, and 0 if any two indices are equal. This antisymmetric symbol is invaluable in expressing cross products and curls within the language of components.

When using the Levi-Civita symbol to define the curl of a vector, one encapsulates the essence of a rotating field through the intricacies of indexed notation. It also becomes the bedrock for proving identities and simplifying expressions that involve cross products, like the curl of a vector product.

When dealing with multivariable functions, the concept of a partial derivative plays a pivotal role. A partial derivative measures how a function changes as one variable changes, keeping other variables constant. In contrast to the ordinary derivative that applies to functions of a single variable, the partial derivative is denoted by \(\partial\) and encapsulates the notion of a directional rate of change.

Calculus becomes multidimensional with partial derivatives. They are the building blocks of gradient, divergence, and curl operations in vector calculus. In our identity, the individual partial derivatives of a scalar function and the components of a vector field lead to a deeper understanding of how scalar and vector fields interact and influence each other.