MULTIVARIABLE CALCULUS 05

The Total Differential

The total differential combines the effects of small changes in all input variables.

The main idea

Suppose

\[ z=f(x,y). \]

If both \(x\) and \(y\) change, then \(z\) may change because of both changes.

For small changes \(dx\) and \(dy\), the corresponding linear change in \(z\) is

\[ \boxed{ dz = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy } \]

This is called the total differential.

1. Why partial derivatives are not enough on their own

The partial derivative

\[ \frac{\partial f}{\partial x} \]

describes change when \(x\) varies and \(y\) is fixed.

Similarly,

\[ \frac{\partial f}{\partial y} \]

describes change when \(y\) varies and \(x\) is fixed.

But if both variables change together, both effects contribute.

2. Contribution from the \(x\)-change

If \(x\) changes by \(dx\), then the linear contribution to \(z\) is

\[ \frac{\partial f}{\partial x}\,dx. \]

This is the local rate of change in the \(x\)-direction multiplied by the small change in \(x\).

3. Contribution from the \(y\)-change

If \(y\) changes by \(dy\), then the linear contribution is

\[ \frac{\partial f}{\partial y}\,dy. \]

Adding the two contributions gives

\[ dz = f_x\,dx+f_y\,dy. \]

4. A simple example

Let

\[ f(x,y)=x^2+y^2. \]

Then

\[ f_x=2x \]

and

\[ f_y=2y. \]

Therefore,

\[ \boxed{ dz = 2x\,dx+2y\,dy } \]

5. Evaluate at a point

\[ (x,y)=(1,2), \]

the total differential becomes

\[ dz = 2\,dx+4\,dy. \]

Suppose

\[ dx=0.1, \qquad dy=0.05. \]

Then

\[ dz = 2(0.1)+4(0.05). \]

Therefore,

\[ \boxed{ dz=0.4 } \]

6. What does \(dz=0.4\) mean?

It means that the tangent-plane approximation predicts that \(z\) will increase by about \(0.4\) when the inputs change by

\[ dx=0.1, \qquad dy=0.05. \]

It is a local linear approximation, not necessarily the exact finite change.

7. Actual change versus differential

The actual finite change is

\[ \boxed{ \Delta z = f(x+\Delta x,y+\Delta y) - f(x,y) } \]

The total differential is

\[ \boxed{ dz = f_x\,dx+f_y\,dy } \]

When the input changes are small and we choose

\[ dx=\Delta x, \qquad dy=\Delta y, \]

we usually have

\[ \boxed{ \Delta z\approx dz } \]

8. Compare exact and approximate change

Take

\[ f(x,y)=x^2+y^2 \]

\[ (x,y)=(1,2). \]

Let

\[ \Delta x=0.1, \qquad \Delta y=0.05. \]

The original value is

\[ f(1,2)=1+4=5. \]

The new value is

\[ f(1.1,2.05) = 1.21+4.2025 = 5.4125. \]

Therefore,

\[ \Delta z = 5.4125-5 = 0.4125. \]

The differential predicted

\[ dz=0.4. \]

\[ \boxed{ \Delta z=0.4125, \qquad dz=0.4 } \]

which are close because the input changes are small.

9. Why the approximation works

Near a point, a smooth surface behaves approximately like its tangent plane.

The total differential gives the change predicted by that tangent plane.

\[ \boxed{ \text{smooth surface} \approx \text{tangent plane locally} } \]

10. Tangent-plane formula

Near the point

\[ (a,b), \]

the function can be approximated by

\[ \boxed{ f(x,y) \approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) } \]

This is the multivariable version of the one-variable linear approximation

\[ f(x+\Delta x) \approx f(x)+f'(x)\Delta x. \]

11. Diagram

12. Vector form

The total differential can be written compactly using the gradient.

Let

\[ d\mathbf{x} = \begin{pmatrix} dx\\ dy \end{pmatrix}. \]

Since

\[ \nabla f = \begin{pmatrix} f_x\\ f_y \end{pmatrix}, \]

we have

\[ \boxed{ df = \nabla f\cdot d\mathbf{x} } \]

So the total differential is the dot product of the gradient with the small input displacement.

13. More than two variables

\[ w=f(x,y,z), \]

then

\[ \boxed{ dw = f_x\,dx + f_y\,dy + f_z\,dz } \]

Each input contributes its own first-order change.

14. Total derivative when variables depend on time

Suppose

\[ z=f(x,y) \]

and both

\[ x=x(t), \qquad y=y(t). \]

Then \(z\) also depends on time.

Divide the total differential by \(dt\):

\[ \frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}. \]

\[ \boxed{ \frac{dz}{dt} = f_x\frac{dx}{dt} + f_y\frac{dy}{dt} } \]

This is the multivariable chain rule.

15. Example of the multivariable chain rule

Let

\[ z=x^2+y^2, \]

where

\[ x=t, \qquad y=t^2. \]

Then

\[ f_x=2x, \qquad f_y=2y. \]

Also,

\[ \frac{dx}{dt}=1, \qquad \frac{dy}{dt}=2t. \]

Therefore,

\[ \frac{dz}{dt} = 2x(1)+2y(2t). \]

Substitute

\[ x=t, \qquad y=t^2. \]

Then

\[ \boxed{ \frac{dz}{dt} = 2t+4t^3 } \]

16. Check by direct substitution

Since

\[ x=t \]

and

\[ y=t^2, \]

we can write

\[ z=t^2+t^4. \]

Differentiate directly:

\[ \frac{dz}{dt} = 2t+4t^3. \]

The same result is obtained.

17. Error and sensitivity

Suppose a measured quantity is

\[ Q=f(x,y). \]

If \(x\) and \(y\) contain small measurement errors \(dx\) and \(dy\), then

\[ dQ = f_x\,dx+f_y\,dy \]

approximates the resulting error in \(Q\).

So total differentials are useful in uncertainty and sensitivity calculations.

18. A modelling interpretation

Suppose

\[ R=R(\beta,\gamma). \]

Then

\[ dR = \frac{\partial R}{\partial\beta}\,d\beta + \frac{\partial R}{\partial\gamma}\,d\gamma. \]

This estimates how a small simultaneous change in both parameters affects \(R\).

19. Partial versus total change

A partial derivative isolates one input:

\[ \frac{\partial f}{\partial x}. \]

The total differential combines all small input changes:

\[ \boxed{ df = \sum_i \frac{\partial f}{\partial x_i}\,dx_i } \]

20. The central idea

\[ \boxed{ \text{small change in }x \Rightarrow f_x\,dx } \] \[ \boxed{ \text{small change in }y \Rightarrow f_y\,dy } \] \[ \boxed{ \text{total small change} = f_x\,dx+f_y\,dy } \]

The total differential is the multivariable version of the one-variable relation

\[ dy=f'(x)\,dx. \]