Partial Differentiation Help with Chain Rule

[Saladsamurai]

In fluid mechanics velocity is given in the form

$$\textbf{V}=u\textbf{i}+v\textbf{j}+w\textbf{k}$$

Homework Statement

A two-dimensional velocity field is given by

$$\textbf{V}=(x^2-y^2+x)\textbf{i}+(-2xy-y)\textbf{j}$$

At $(x_o,y_o)$ compute the accelerations $a_x\text{ and }a_y$

I am having trouble with the books definition of $a_x\text{ and }a_y$

Now I can see that the w term is zero. They define a_x to be

$$a_x=\frac{\partial{u}}{\partial{t}}=u\frac{\partial{u}}{\partial{x}}+v\frac{\partial{u}}{\partial{y}}$$

I know that the chain rule has been used to arrive at this, but I seem to be getting lost along the way. So I am attempting to do it out here. So when we say

$\frac{\partial{u}}{\partial{t}}$ we are really saying

$$\frac{\partial{}}{\partial{t}}[u(x(t),y(t),z(t))]$$

right?

I am just confused as to how to evaluate this

 

[slider142]

That is a bit strange, seeing as u is only a function of one variable, t. I don't see why the partial symbol is being used, as it is thus equivalent to the normal derivative du/dt. Since u(t) = u(x(t), y(t), z(t)) is a map from R into R, we have:
$$\frac{du}{dt} = \frac{\partial u}{\partial x}\frac{dx}{dt} + \frac{\partial u}{\partial y}\frac{dy}{dt} + \frac{\partial u}{\partial z}\frac{dz}{dt}$$
by the chain rule.

Now I can see that the w term is zero. They define a_x to be
$$a_x=\frac{\partial{u}}{\partial{t}}=u\frac{\partia l{u}}{\partial{x}}+v\frac{\partial{u}}{\partial{y} }$$

I do not see any way in which they can justify v being part of the partial derivative of u with respect to time.

 

[Saladsamurai]

I do not see any way in which they can justify v being part of the partial derivative of u with respect to time.

I guess I should have stated that u,v,w are ALL functions of x,y,z,t (or can be)

So u could be a function of x, y, & z each of which are functions of time...I can see where this is going, but I do not know how to get there. Unfortunately there are some "gaps" in my fundamental knowledge of calculus. They are mostly notational in nature.

I know that

$$\frac{\partial{}}{\partial{t}}[u(x(t),y(t),z(t))]$$



$$=\frac{\partial{u}}{\partial{x}}*\frac{\partial{x}}{\partial{t}}+\frac{\partial{u}}{\partial{y}}*\frac{\partial{y}}{\partial{t}}+\frac{\partial{u}}{\partial{z}}*\frac{\partial{z}}{\partial{t}}$$

but I am not sure how to get from that first line to the next.

Seriously, I am just confused by the commas. :rofl:

 

[slider142]

I guess I should have stated that u,v,w are ALL functions of x,y,z,t (or can be)

So u could be a function of x, y, & z each of which are functions of time...I can see where this is going, but I do not know how to get there. Unfortunately there are some "gaps" in my fundamental knowledge of calculus. They are mostly notational in nature.

I know that

$$\frac{\partial{}}{\partial{t}}[u(x(t),y(t),z(t))]$$



$$=\frac{\partial{u}}{\partial{x}}*\frac{\partial{x}}{\partial{t}}+\frac{\partial{u}}{\partial{y}}*\frac{\partial{y}}{\partial{t}}+\frac{\partial{u}}{\partial{z}}*\frac{\partial{z}}{\partial{t}}$$

but I am not sure how to get from that first line to the next.

Seriously, I am just confused by the commas. :rofl:

Oh, never mind. they're using the fact that
$$u(t) = \frac{dx}{dt}$$
and
$$v(t) = \frac{dy}{dt}$$
which goes right into the chain rule we already posted.

 

[Saladsamurai]

I was actually being serious about the whole "being confused by the commas" thing.

How do we actually "operate on" $[u(x(t),y(t),z(t))]$ ? That is, what are the mechanical steps to get from

$$\frac{\partial{}}{\partial{t}}[u(x(t),y(t),z(t))]$$

to

$$=\frac{\partial{u}}{\partial{x}}*\frac{\partial{x} }{\partial{t}}+\frac{\partial{u}}{\partial{y}}*\frac{\partial{y}}{\partial{t}}+\frac{\partial{u}}{\partial{z}}*\frac{\partial{z}}{\partial{t}}$$

 

[Fredrik]

There are no intermediate steps. That is the chain rule.

I like to write the single-variable version of it in the form

$$(f\circ g)'(x)=f'(g(x))g'(x)$$

and the many-variables version of it in the form

$$(f\circ g)^i{}_{,j}(x)=f^i{}_{,k}(g(x))g^k{}_{,j}(x)$$

because this form is fairly easy to remember. Repeated indices are summed over (Einstein's summation convention), and a ",j" denotes partial derivation with respect to variable number j. For example, $g^k{}_{,j}(x)$ is the partial derivative with respect to the jth variable of the kth component of the function g.

In your case, $g:\mathbb R\rightarrow\mathbb R^3,\ f:\mathbb R^3\rightarrow\mathbb R$, so the above reduces to

$$(f\circ g)'(x)=f_{,k}(g(x))g^k'(x)$$

 

[Saladsamurai]

Wow. Engineers really get the short end of the stick when it comes to thoroughness in their maths. I have been thinking about reteaching myself Calculus from a more 'Pure' standpoint.

Do you have any suggestions on a good textbook for someone who already knows how to 'do' calculus problems (for the most part) but really wants to understand the 'hows and whys' of it all?

 

[Fredrik]

Theorems like the chain rule are proved in all the standard textbooks. I'm sure you can find recommendations in the science book forum. (The books I studied are in Swedish, so you probably wouldn't like them). If you want a book that's just about definitions and theorems, I can recommend "Principles of mathematical analysis" by Walter Rudin. I should warn you though. It's a pretty difficult subject.

 

[slider142]

Wow. Engineers really get the short end of the stick when it comes to thoroughness in their maths. I have been thinking about reteaching myself Calculus from a more 'Pure' standpoint.

Do you have any suggestions on a good textbook for someone who already knows how to 'do' calculus problems (for the most part) but really wants to understand the 'hows and whys' of it all?

The commas are just there to denote a map from R into R³.
Spivak's "Calculus on Manifolds" is the definitive text on the topic. It is concise and easily followable by anyone who has taken single-variable analysis and has exposure to multivariable calculus. The proofs are either left on exercises The following two theorems are proven in the chapter on differentiation and can be used to differentiate any function using the chain rule (the notation is Euler's notation, $D_if^j$ refers to the partial derivative with respect to the ith variable of the jth component of f):
This theorem allows you to find the derivative of vector-valued functions in matrix form.
Theorem: If $f:R^n\rightarrow R^m$ is differentiable at a, then $D_jf^i(a)$ exists for $1\leq i\leq m, 1\leq j\leq n$ and f'(a) is the $m\times n$ matrix $(D_jf^i(a))$.
This theorem is the multivariable chain rule.
Theorem:If $g_1,\cdots,g_m:R^n\rightarrow R$ are continuously differentiable at a, and $f:R^m\rightarrow R$ is differentiable at $(g_1(a),\cdots,g_m(a))$, then $F:R^n\rightarrow R$ defined by $F(x) = f(g_1(x),\cdots,g_m(x))$ is also differentiable at a and
$$D_iF(a) = \sum_{j=1}^m D_jf(g_1(a),\cdots,g_m(a)) D_ig_j(a).$$
Spivak makes the proofs of these theorems very easy by using very good definitions and leading the reader in the right directions with the exercises so that the reader can usually prove the theorem themselves without Spivak's proof. Ie., Stokes's Theorem, a crown jewel of vector calculus, simply falls out of the definitions by rote algebra.

 

[Saladsamurai]

The commas are just there to denote a map from R into R³.
Spivak's "Calculus on Manifolds" is the definitive text on the topic. It is concise and easily followable by anyone who has taken single-variable analysis and has exposure to multivariable calculus. The proofs are either left on exercises The following two theorems are proven in the chapter on differentiation and can be used to differentiate any function using the chain rule (the notation is Euler's notation, $D_if^j$ refers to the partial derivative with respect to the ith variable of the jth component of f):
This theorem allows you to find the derivative of vector-valued functions in matrix form.
Theorem: If $f:R^n\rightarrow R^m$ is differentiable at a, then $D_jf^i(a)$ exists for $1\leq i\leq m, 1\leq j\leq n$ and f'(a) is the $m\times n$ matrix $(D_jf^i(a))$.
This theorem is the multivariable chain rule.
Theorem:If $g_1,\cdots,g_m:R^n\rightarrow R$ are continuously differentiable at a, and $f:R^m\rightarrow R$ is differentiable at $(g_1(a),\cdots,g_m(a))$, then $F:R^n\rightarrow R$ defined by $F(x) = f(g_1(x),\cdots,g_m(x))$ is also differentiable at a and
$$D_iF(a) = \sum_{j=1}^m D_jf(g_1(a),\cdots,g_m(a)) D_ig_j(a).$$
Spivak makes the proofs of these theorems very easy by using very good definitions and leading the reader in the right directions with the exercises so that the reader can usually prove the theorem themselves without Spivak's proof. Ie., Stokes's Theorem, a crown jewel of vector calculus, simply falls out of the definitions by rote algebra.

I have taken single-var & multi-var calculus, but no Analysis. What would you suggest? I don't even know what a Map is and I have never really used the R³ notations.

 

[slider142]

In that case, you can use a "baby analysis" text to bridge the gap. That is, a text that aims to teach calculus with rigor, covering the properties of the real number field, limits, series, derivatives, and integrals with complete rigor and every important theorem either proven explicitly by the author or the student being guided to the proof by exercises. There is no analogical hand-waving, appeals to physical situations as a substitute for proof and no references to other texts for the proof (ie., this proof is too involved to be covered here...). while there are many texts that do this, there are a few that have become classics for every generation by being extremely well-written and actually enjoyable to read and do.
https://www.amazon.com/dp/0471588814/?tag=pfamazon01-20 are the most popular. Once you have picked up and read at least the first chapter of any of these, you will see why they're so popular. They're almost like having a highly knowledgeable friendly professor in the room with you.
In order to understand the aim of multivariable and vector calculus, you will have to have a good grounding in linear algebra, for which I recommend https://www.amazon.com/dp/0387982582/?tag=pfamazon01-20, supplemented by a matrix-based algebra text if you so choose, or a Schaum's Outline for numerical exercises. Axler's text is more modern algebra and geometry oriented than most numerically oriented matrix algebra texts, which is more important for the intuitive application to calculus, which, in vector form, pretty much tries to study nonlinear objects by reducing them to locally linear objects which we study with the well-established tools of linear algebra.