What is an intuition behind total differential in two variables function?

Question

As the definition, the total differential of a differentiable function with two variables equal to: $$ dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy $$ Since there are innumerable derivable directions, I confuse it now. I have two confusion in follow:

1. Why the total differential equal to sum of just two partial differentials?
2. For a differentiable function, the total differential equal to sum of any two different direction's partial differentials?

Answer 1

I prefer to start from a definition of the total derivative that isn’t tied to a specific coordinate system. If $f:\mathbb R^m\to\mathbb R^n$, it is differentiable at $\mathbf v\in\mathbb R^m$ if there is a linear map $L_{\mathbf v}:\mathbb R^m\to\mathbb R^n$ such that $f(\mathbf v+\mathbf h)=f(\mathbf v)+L_{\mathbf v}[\mathbf h]+o(\|\mathbf h\|)$. The linear map $L_{\mathbf v}$ is called the differential or total derivative of $f$ at $\mathbf v$, denoted by $\mathrm df_{\mathbf v}$ or simply $\mathrm df$. The idea here is that $\mathrm df_{\mathbf v}$ is the best linear approximation to the change in $f$ near $\mathbf v$, with the error of this approximation vanishing “faster” than the displacement $\mathbf h$.

Relative to some specific pair of bases for the domain and range of $f$, $\mathrm df$ can be represented by an $n\times m$ matrix. To see what this matrix is, you can treat $f$ as a vector of functions:$$f(\mathbf v)=\pmatrix{\phi_1(\mathbf v)\\\phi_2(\mathbf v)\\\vdots\\\phi_n(\mathbf v)}$$ or, written in terms of coordinates, $$\begin{align}y_1&=\phi_1(x_1,x_2,\dots,x_m)\\y_2&=\phi_2(x_1,x_2,\dots,x_m)\\\vdots\\y_n&=\phi_n(x_1,x_2,\dots,x_m).\end{align}$$ The matrix of $\mathrm df$ then turns out to be the Jacobian matrix of partial derivatives $$\pmatrix{{\partial\phi_1\over\partial x_1}&{\partial\phi_1\over\partial x_2}&\cdots&{\partial\phi_1\over\partial x_m}\\{\partial\phi_2\over\partial x_1}&{\partial\phi_2\over\partial x_2}&\cdots&{\partial\phi_2\over\partial x_m}\\\vdots&\vdots&\ddots&\vdots\\{\partial\phi_n\over\partial x_1}&{\partial\phi_n\over\partial x_2}&\cdots&{\partial\phi_n\over\partial x_m}}.$$ The displacement vector $\mathbf h$ can be written as $\mathrm d\mathbf v=(\mathrm dx^1,\mathrm dx^2,\dots,\mathrm dx^m)^T$. (The $\mathrm dx^i$ here can themselves be thought of as differentials of affine coordinate functions, but that’s not an important detail for this discussion.)

For the special case of a scalar function $f:\mathbb R^m\to\mathbb R$, $\mathrm df[\mathbf h]$ becomes $${\partial f\over\partial x_1}\mathrm dx^1+{\partial f\over\partial x_2}\mathrm dx^2+\cdots+{\partial f\over\partial x_m}\mathrm dx^m.$$ Now, the partial derivative ${\partial f\over\partial x_i}$ is just the directional derivative of $f$ in the direction of the $x^i$-axis, so this formula expresses the total derivative of $f$ in terms of its directional derivatives in a particular set of directions. Notice that there was nothing special about the basis we chose for $\mathbb R^m$. If we choose a different basis, $\mathrm df$ will have the same form, but the derivatives will be taken in a different set of directions. In your case of $\mathbb R^2$, a basis consists of two vectors, so derivatives in only two directions are sufficient to completely specify the total derivative. If you understand it as a linear map from $\mathbb R^2$ to $\mathbb R$, this should come as no surprise.

Answer 2

Suppose I have a scalar field on the plane given by the formula

$$ s = x + y^2 + e^r + \sin(\theta) $$

Yes, this formula for $s$ mixes both cartesian and polar coordinates on the plane!

Using this formula, we can compute the total differential to be

$$ \mathrm{d}s = \mathrm{d}x + 2 y \,\mathrm{d}y + e^r \,\mathrm{d}r + \cos(\theta) \,\mathrm{d}\theta $$

So don't think of it as doing some calculation with just the right number of partial derivatives — think of it as just the extension of the familiar methods of computing derivatives. Partial derivatives only enter the picture when you are specifically interested in computing the differential of a function that has more than one argument; e.g. to compute $\mathrm{d}f(s,t)$ for some function $f$ of two arguments.

Of course, we can rewrite; e.g. it using equations like $\mathrm{d}x = \mathrm{d}(r \cos(\theta)) = \cos(\theta)\, \mathrm{d}r - r \sin(\theta) \, \mathrm{d}\theta$ and $\mathrm{d}y = \sin(\theta) \,\mathrm{d}r + r \cos(\theta) \, \mathrm{d}\theta$ to get rid of the $\mathrm{d}x$ and $\mathrm{d}y$ terms and leaving the result in terms of $\mathrm{d}r$ and $\mathrm{d}\theta$.

In the plane, there are only two independent differentials, so we can always rewrite as a linear combination of two of them.

In my opinion, the better way to think about things is that the total differential is the most natural form of the derivative, and the partial derivative is a linear functional on differential forms; e.g. in the standard $x-y$ coordates, $\partial/\partial x$ is the mapping that sends $\mathrm{d}x \to 1$ and $\mathrm{d}y \to 0$.

So, using the notation $\partial z / \partial x$ for the action of $\partial / \partial x$ on $\mathrm{d}z$, we see that if we have an equation

$$ \mathrm{d}z = f \,\mathrm{d}x + g \,\mathrm{d}y $$

then

$$ \frac{\partial z}{\partial x} = f \cdot 1 + g \cdot 0 = f$$ $$ \frac{\partial z}{\partial y} = f \cdot 0 + g \cdot 1 = g$$

and so we'd have

$$ \mathrm{d}z = \frac{\partial z}{\partial x}\, \mathrm{d}x + \frac{\partial z}{\partial y} \,\mathrm{d} y$$

Aside: another advantage the total differential has over the partial derivative is that it's actually self-contained. In the plane, $\partial / \partial x$ has no meaning on its own; e.g. if we set $w=x+y$, then $\partial / \partial x$ means something different when expressing things as a function of $(x,y)$ than it does when expressing things as a function of $(x,w)$. (in the former, it sends $\mathrm{d}y \to 0$, in the latter it sends $\mathrm{d}w \to 0$, and thus $\mathrm{d}y \to -1$).

Answer 3

1. It's not that $dz$ contains just two directional derivative. Instead, you may think of $$ dz = \frac{\partial z}{\partial x } dx + \frac{\partial z}{\partial y }dy$$ as representating $dz$ under the basis $\{dx, dy\}$. Let $v = (v_1, v_2)$ be a vector, then $$dz (v) = \frac{\partial z}{\partial x } dx (v)+ \frac{\partial z}{\partial y }dy(v) = \frac{\partial z}{\partial x } v_1 + \frac{\partial z}{\partial y }v_2 = D_v z,$$ where $D_v z$ is the directional derivative of $z$ along the direction $v$. So all directional derivative of $z$ is encoded in the formula already.

2. Almost true. Let $v, w$ be two independent vectors. Let $v^*, w^*$ be the dual vector defined by $$\tag{1} v^*(av + bw) = a,\ \ \ w^*(av + bw) =b,\ \ \ \forall a, b \in \mathbb R.$$ From $(1)$, we can represent $v, w$ using the basis $\{dx,dy\}$: indeed if we write $$\tag{2} \begin{pmatrix}v^* \\ w^* \end{pmatrix} = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \begin{pmatrix}dx \\ dy \end{pmatrix},$$ one sees that $$\tag{3} \begin{pmatrix}A & B\\ C & D \end{pmatrix} \begin{pmatrix}v_1 & w_1\\ v_2 & w_2 \end{pmatrix}= \begin{pmatrix}1 & 0\\ 0 & 1 \end{pmatrix}, \Rightarrow \begin{pmatrix}A & B\\ C & D \end{pmatrix} =\begin{pmatrix}v_1 & w_1\\ v_2 & w_2 \end{pmatrix}^{-1} $$ (we need that $\{v, w\}$ be linear independent so that the right hand side is defined). Thus we have, using $(2)$ and $(3)$, $$\begin{split} (D_v z) v^* + (D_w z) w^* &= \begin{pmatrix} D_v z & D_wz\end{pmatrix} \begin{pmatrix} v^* \\ w^*\end{pmatrix} \\ &= \begin{pmatrix} v_1\frac{\partial z}{\partial x} + v_2 \frac{\partial z}{\partial y} & w_1 \frac{\partial z}{\partial x} + w_2 \frac{\partial z}{\partial y}\end{pmatrix} \begin{pmatrix} v^* \\ w^*\end{pmatrix} \\ &= \begin{pmatrix} \frac{\partial z}{\partial x} & \frac{\partial z}{\partial y}\end{pmatrix} \begin{pmatrix} v_1 & w_1 \\ v_2 & w_2 \end{pmatrix}\begin{pmatrix} v^* \\ w^*\end{pmatrix}\\ &= \begin{pmatrix} \frac{\partial z}{\partial x} & \frac{\partial z}{\partial y}\end{pmatrix} \begin{pmatrix} dx \\ dy\end{pmatrix} \\ &=\frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y}dy = dz \end{split}$$ Thus it is true that you can use any two independent vectors to represent $dz$ as $$dz = (D_v z) v^* + (D_wz) w^*.$$

Answer 4

Any infinitely small change in $(x,y)$ includes a change $dx$ in $x$ and a change $dy$ in $y$. The resulting change in $z$ resulting from the change in $x$ is $\dfrac{\partial z}{\partial x} \, dx$, and to that we add a change in $z$ resulting from the change in $y$.

Answer 5

I used to be confused about this also, until I learned that $(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y})$ is a vector-like object.

What it means is that you can rotate the coordinate axes so that they point in any direction and rewrite the total derivative in terms of the partial derivatives along those axes, and under such a rotation the partial derivatives and the displacements dx and dy transform in such a way that leaves the total derivative unchanged.

It was Feynman's explanation in Volume II Chapter 2 that cleared up my confusion.

Answer 6

I know this post is old but I thought it might be useful to share my understanding of differentials since I'm not a mathematician and thus it might be more relatable to folks like me that visited this questions. My understanding is through linear approximations. This is probably an easier way to acquire some intuition about it. I will start with a 2D case first and then jump to the 3D one.

Let’s say we a function f(x) (e.g. y = x ² + 1) and a point P(a, b) that is part of it. Therefore, the equation for a tangent line to f(x) at P(a, b) is:

[1] y - b = f’(a)(x - a) (point-slope form)

where f’(a) is the derivative of f(x) when x = a. Therefore at P(a, b)there is a tangent line L(x) = f’(a)(x - a) + b

Now the differential dy is defined as:

[2] dy = f’(x)dx

You can now see the similarity between eq. [1] and [2]. Here dy = y - b which is the change in the value of y as we go from x = a to some arbitrary value x (a + Δx = a + dx) on the tangent line L(x). Noticed that from a mathematical point a view dx and dy can have the value of any real number. But if dx is sufficiently small then dy ~ Δy, that is, f(a + dx) ~ f(a) + dy; where Δy is the change of our function f(x). Differentials were defined here in terms of changes and a linear approximation of a function.

Analogous to the tangent line you can have a tangent plane for the 3D case. Let’s say we have a surface f(x,y) (e.g. z = x² + y²) and a point on it P(a,b,c). At P(a,b,c) you have two tangent lines, one in the x direction and one in the y direction. Therefore, the slopes of the two tangent lines will be the partial derivatives of f(x,y): f_x(x,y) and f_y(x,y) at P(a,b,c), respectively. Because the tangent plane at P(a,b,c) will contain the two tangent lines, we can derive the equation of the tangent plane T(x,y) at P(a,b,c) to be:

[3] z – c = fx(a, b)(x – a) + fy(a, b)(y – b)

or T(x, y) = f_x(a, b)(x – a) + f_y(a, b)(y – b) + c

The total differential is defined as:

[4] dz = f_x(x, y)dx + f_y(x,y)dy

Compare [3] and [4] and you will see that dz is the change of the T(x,y) as we go from points P(a,b,c) to P(x,y,z). If dx and dy are sufficiently small then dz ~ Δz: it approximates the change of our function f(x,y) as we move from P(a,b,c) by dx and dy.

Hope that helps!

Mike