Differential Calculus: From Basics to Advanced | bAstronautWay

Advanced Mathematics

Table of Contents

Differential
Calculus

Master partial differentiation, chain rule, total derivatives, Euler’s theorem, Jacobians and functional dependence through concepts, derivations and worked mathematical examples.

Presented by Grandmaster Bikram Sutradhar
bAstronautWay

Complete Syllabus

✓ Partial Differentiation
✓ Chain Rule
✓ Change of Variables
✓ Total Derivative
✓ Mixed Partial Derivatives
✓ Euler’s Theorem
✓ Converse of Euler’s Theorem
✓ Jacobian
✓ Functional Dependence
✓ MCQ Examination

Learning Roadmap

What You Will Learn

Partial Differentiation

Differentiate multivariable functions while keeping other variables constant.

Chain Rule

Differentiate composite functions involving several dependent variables.

Change of Variables

Transform derivatives when variables are replaced by new coordinates.

Total Derivative

Understand how all changing variables contribute to a differential.

Mixed Partials

Understand conditions under which mixed partial derivatives are equal.

Euler’s Theorem

Use homogeneity to obtain powerful differential identities.

Jacobian

Study determinant-based transformation of multivariable variables.

Functional Dependence

Determine whether several functions are independently variable.

1. Partial Differentiation

Suppose a function depends on two or more independent variables:

z = f(x,y)

The partial derivative with respect to x means that y is treated as constant.

∂z/∂x = fₓ

Similarly, while differentiating with respect to y, x is kept constant:

∂z/∂y = fᵧ

Example 1

z = x³y² + 4xy + y³

With respect to x:

∂z/∂x = 3x²y² + 4y

With respect to y:

∂z/∂y = 2x³y + 4x + 3y²

Golden Rule: For ∂/∂x, treat every other independent variable as a constant.

Second Partial Derivatives

fₓₓ = ∂²f/∂x²

fᵧᵧ = ∂²f/∂y²

fₓᵧ = ∂/∂y(∂f/∂x)

fᵧₓ = ∂/∂x(∂f/∂y)

2. Chain Rule for Partial Derivatives

The chain rule is used when a variable depends indirectly on other variables.

Suppose

z=f(x,y)

and

x=x(t), y=y(t)

Then

dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)

Example

z=x²+y²

where

x=t², y=3t

First:

∂z/∂x=2x
∂z/∂y=2y

Also:

dx/dt=2t
dy/dt=3

Therefore:

dz/dt = 2x(2t)+2y(3)

Substitute x=t² and y=3t:

dz/dt = 4t³+18t

3. Change of Variables

Suppose x and y are functions of u and v:

x=x(u,v)
y=y(u,v)

and

z=f(x,y)

Then:

∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u)

Similarly:

∂z/∂v = (∂z/∂x)(∂x/∂v) + (∂z/∂y)(∂y/∂v)

Example

Let

z=x²+y²

and

x=u+v
y=u−v

Then

z=(u+v)²+(u−v)²

Therefore

z=2u²+2v²

Hence

∂z/∂u=4u
∂z/∂v=4v

Changing variables is especially important in polar, cylindrical, spherical and other coordinate transformations.

4. Total Derivative

If z depends on x and y and both x and y depend on another variable, then the change in z contains contributions from both x and y.

z=f(x,y)

then the total differential is

dz = (∂z/∂x)dx + (∂z/∂y)dy

If x=x(t) and y=y(t):

dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)

Three Variables

u=f(x,y,z)

then

du = uₓ dx + uᵧ dy + u_z dz

If x,y,z depend on t:

du/dt = uₓ dx/dt + uᵧ dy/dt + u_z dz/dt

Do not confuse:
Partial derivative = vary one independent variable while keeping others fixed.

Total derivative = account for all variables that are changing.

5. Equality of Mixed Partial Derivatives

For a sufficiently smooth function, under the usual continuity conditions on the relevant second partial derivatives:

∂²f/∂x∂y = ∂²f/∂y∂x

This is commonly associated with Clairaut’s theorem (or Schwarz’s theorem).

Example

f=x³y²+4xy

First differentiate with respect to x:

fₓ=3x²y²+4y

Now y:

fₓᵧ=6x²y+4

Reverse order:

fᵧ=2x³y+4x

Then x:

fᵧₓ=6x²y+4

Therefore:

fₓᵧ=fᵧₓ

Exam shortcut: For polynomial functions, mixed partials are normally straightforward to calculate and will agree.

6. Homogeneous Functions of Two Variables

A function f(x,y) is homogeneous of degree n if:

f(λx,λy)=λⁿf(x,y)

for a suitable scaling parameter λ.

Example

f(x,y)=x³+2x²y+xy²

Every term has total degree 3.Therefore f is homogeneous of degree 3.

f(λx,λy) = λ³f(x,y)

7. Euler’s Theorem for Homogeneous Functions

If f(x,y) is homogeneous of degree n, then:

x(∂f/∂x)+y(∂f/∂y)=nf

This is one of the most important formulas in this chapter.

Proof

Since f is homogeneous of degree n:

f(tx,ty)=tⁿf(x,y)

Differentiate both sides with respect to t:

fₓ(tx,ty)x + fᵧ(tx,ty)y = ntⁿ⁻¹f(x,y)

Put t=1:

xfₓ+yfᵧ=nf

Hence proved.

Example

f=x³+2x²y+xy²

Degree:

n=3

Compute:

fₓ=3x²+4xy+y²

fᵧ=2x²+2xy

Therefore:

xfₓ+yfᵧ = 3x³+6x²y+3xy²

which is

Thus Euler’s theorem is verified.

8. Euler’s Theorem for Three Variables

u=f(x,y,z)

is homogeneous of degree n, then:

x uₓ + y uᵧ + z u_z = nu

Example

u=x²y+y²z+z²x

Each term has degree 3.Therefore n=3.Euler’s theorem gives:

xuₓ+yuᵧ+zu_z=3u

Memory Formula:

2 variables → x fₓ + y fᵧ = nf

3 variables → x fₓ + y fᵧ + z f_z = nf

9. Second-Order Euler Relation

If f(x,y) is homogeneous of degree n, differentiating Euler’s identity gives a useful second-order relation:

x²fₓₓ + 2xyfₓᵧ + y²fᵧᵧ = n(n−1)f

For three variables:

x²uₓₓ+y²uᵧᵧ+z²uzz +2xyuₓᵧ +2yzuᵧz +2zxu_zₓ = n(n−1)u

This second-order form is extremely useful in higher-level exam problems involving homogeneous functions.

10. Converse of Euler’s Theorem

Under appropriate differentiability assumptions, if a differentiable function satisfies

xfₓ+yfᵧ=nf

throughout a suitable domain, then f behaves as a homogeneous function of degree n.In other words, Euler’s differential identity can be used as a test for homogeneity.

Example

Suppose

f=x²+y²

Then

fₓ=2x
fᵧ=2y

Hence

xfₓ+yfᵧ = 2x²+2y² = 2f

Therefore f is homogeneous of degree 2.

Exam Strategy: If you are given a differential equation of the form xfₓ+yfᵧ=nf, immediately suspect homogeneity of degree n.

11. Jacobian

The Jacobian describes how a transformation between variables changes locally.

For

u=u(x,y)
v=v(x,y)

the Jacobian is

∂(u,v)/∂(x,y) = | uₓ uᵧ | | vₓ vᵧ |

Therefore:

∂(u,v)/∂(x,y) = uₓvᵧ-uᵧvₓ

Example

Let

u=x+y
v=x−y

Then

uₓ=1, uᵧ=1
vₓ=1, vᵧ=−1

Hence

J = (1)(−1)−(1)(1) = −2

Therefore:

∂(u,v)/∂(x,y)=−2

12. Reciprocal Property of Jacobians

When the transformation is locally invertible and the required derivatives exist:

∂(u,v)/∂(x,y) × ∂(x,y)/∂(u,v) = 1

Therefore:

∂(x,y)/∂(u,v) = 1 / [∂(u,v)/∂(x,y)]

provided the Jacobian is non-zero.

If the Jacobian is zero, the simple reciprocal formula cannot be used at that point as an ordinary local inverse formula.

13. Functional Dependence

Functions u and v are functionally dependent if there exists a non-trivial relation:

F(u,v)=0

between them.For two functions of two variables, a standard test is:

∂(u,v)/∂(x,y)=0

If the Jacobian vanishes identically in the relevant region, the functions are functionally dependent under the usual regularity assumptions.

Example

Let

u=x+y
v=(x+y)²

Clearly:

v=u²

Therefore:

v−u²=0

So u and v are functionally dependent.Check the Jacobian:

uₓ=1, uᵧ=1
vₓ=2(x+y), vᵧ=2(x+y)

Therefore:

J = 1·2(x+y) − 1·2(x+y) = 0

Hence the Jacobian test confirms the dependence.

⚡ Master Formula Sheet

Concept	Formula
Partial derivative	fₓ = ∂f/∂x
Total differential	dz=fₓdx+fᵧdy
Chain rule	dz/dt=fₓ dx/dt+fᵧ dy/dt
Mixed partials	fₓᵧ=fᵧₓ under suitable regularity conditions
Homogeneity	f(λx,λy)=λⁿf(x,y)
Euler: 2 variables	xfₓ+yfᵧ=nf
Euler: 3 variables	xfₓ+yfᵧ+zf_z=nf
Second Euler relation	x²fₓₓ+2xyfₓᵧ+y²fᵧᵧ=n(n−1)f
Jacobian	∂(u,v)/∂(x,y)=uₓvᵧ−uᵧvₓ
Reciprocal Jacobian	∂(u,v)/∂(x,y) · ∂(x,y)/∂(u,v)=1
Functional dependence	J=0 under appropriate conditions

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Differential Calculus

Complete Syllabus

What You Will Learn

Partial Differentiation

Chain Rule

Change of Variables

Total Derivative

Mixed Partials

Euler’s Theorem

Jacobian

Functional Dependence

1. Partial Differentiation

Example 1

Second Partial Derivatives

2. Chain Rule for Partial Derivatives

Example

3. Change of Variables

Example

4. Total Derivative

Three Variables

5. Equality of Mixed Partial Derivatives

Example

6. Homogeneous Functions of Two Variables

Example

7. Euler’s Theorem for Homogeneous Functions

Proof

Example

8. Euler’s Theorem for Three Variables

Example

9. Second-Order Euler Relation

10. Converse of Euler’s Theorem

Example

11. Jacobian

Example

12. Reciprocal Property of Jacobians

13. Functional Dependence

Example

⚡ Master Formula Sheet