# Math methods in physics book - vector calc proof

• galactic
In summary, the conversation discusses the del operator identity in vector calculus and the various ways it can be written. The use of partial derivatives and the baccab rule are also mentioned. The conversation also includes a request for help in understanding the first few steps of the proof and a suggestion to look up vector calculus identities on Wikipedia.
galactic
Hi all, I just got mary boas math methods in physics book as a supplement because I'm a physics major and I'm browsing thru the vector calculus sections and came across the del operator identity:

nambla (V dot U) = stuff

nambla is the del operator and "dot" is dot product...

I'm trying to figure out how to prove this seeing as I'm very rusty on my kronecker delta, levi-civita permutation tensor, and other vector calc related identities

any tips on the first couple steps?

the solution is on wikipedia if you google "vector calc identities" and it appears that it involves two partial derivative product rules or something

Anyway, I've been bored and stuck on what to do with this for a while tonight and can't figure out how to get the first few steps done that would very much refresh my brain ! many thanks to anyone who is willing to help

Well the natural thing to do is write

$$\mathbf{\nabla(a \cdot b)=\nabla_a (a \cdot b)+\nabla_b (a \cdot b)}$$

where the subscript means the derivative is partial and only effects one vector. However partial vector derivatives are hard to interpret as they represent an arbitrary and artificial separation so we employ the identity

$$\mathbf{\nabla_b (a \cdot b)=(a\times\nabla)\times b+a \, (\nabla\cdot b)=a\times(\nabla\times b)+(a\cdot\nabla)b}$$

The third version being the one used in most books. This illustrates several peculiar things. We have three ways of writing the same thing, yet when switching between them we can be confused. Some object to the partial derivative formulation when the others forms are the same anyway. The fact that we prefer to use right acting operators breaks symmetry (observe the two forms are right and left hand versions of the same thing). Also notice that these rules follow from the so called baccab rule

$$\mathbf{a \times (b \times c)=b \, (a \cdot c)- c \, (a \cdot b)}$$

You can write all this out with your beloved epsilons and deltas if you like. The baccab rule is then written

$$\epsilon_{ijk} \epsilon^{imn}=\left| \begin{array}{} \delta_j^m & \delta_j^n \\ \delta_k^m & \delta_k^n \\ \end{array} \right|=\delta_j^m\delta_k^n-\delta_j^n\delta_k^m$$

What is your native language? I am used to "nabla", not "nambla" but it might be a difference in language. In any case, we think of "nabla", $\nabla$ as the "vector differential operator",$(\partial/\partial x)\vec{i}+ (\partial/\partial y)\vec{j}+ (\partial/\partial z)\vec{k})$. Applied to a scalar valued function f(x) that gives the vector function $(\partial f/\partial x)\vec{i}+ (\partial f/\partial y)\vec{j}+ (\partial f/\partial z)\vec{k})$. If f is the result of a dot product, $f= \vec{u}\cdot\vec{v}= u_xv_x+ u_yv_y+ u_zv_z$ then that formula becomes $(\partial (u_xv_x+ u_yv_y+ u_zv_z)/\partial x)\vec{i}+ (\partial (u_xv_x+ u_yv_y+ u_zv_z)/\partial y)\vec{j}+ (\partial (u_xv_x+ u_yv_y+ u_zv_z)/\partial z)\vec{k})$.

Applying the sum and product rules to those (tedious, so I am not going to do it, but doable) gives the formula.

## 1. What is the purpose of a "Math methods in physics" book?

The purpose of a "Math methods in physics" book is to provide a comprehensive guide to the mathematical techniques and tools used in physics. It aims to help readers develop a solid understanding of the mathematical foundations of physics and apply them to solve problems in various areas of physics.

## 2. What is vector calculus and why is it important in physics?

Vector calculus is a branch of mathematics that deals with the study of vectors and vector fields. It is important in physics because many physical quantities, such as velocity, acceleration, and force, are represented by vectors. Vector calculus allows us to perform mathematical operations on these vectors, such as differentiation and integration, which are essential for solving physics problems.

## 3. What are some common proofs related to vector calculus in a "Math methods in physics" book?

Some common proofs related to vector calculus in a "Math methods in physics" book include proving the divergence theorem, the gradient theorem, and the curl theorem. These theorems are fundamental concepts in vector calculus and are often used in the derivation of important equations in physics, such as Maxwell's equations.

## 4. How can a "Math methods in physics" book help me in my physics studies?

A "Math methods in physics" book can help you in your physics studies by providing a solid foundation in the mathematical concepts and techniques used in physics. It can also help you understand the derivations of important equations and theorems, and how they are applied in solving physics problems. Additionally, the book may include practice problems and exercises to further enhance your understanding and skills.

## 5. Is prior knowledge of advanced mathematics required to understand a "Math methods in physics" book?

While some prior knowledge of mathematics, such as calculus and linear algebra, may be helpful, it is not necessarily required to understand a "Math methods in physics" book. These books are designed to introduce readers to the mathematical concepts and techniques used in physics, starting from the basics. However, a strong foundation in mathematics can make it easier to grasp the concepts presented in the book.

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