LAC Unit 1: Partial Differentiation & Multivariable Calculus

LAC Unit 1 — Partial Differentiation & Multivariable Calculus

Course: Linear Algebra and Calculus
Course Code: 25APS0101
University: Chitkara University, Punjab • Semester 01
Prescribed Textbooks: B.V. Ramana: Higher Engineering Mathematics (McGraw-Hill); Erwin Kreyszig: Advanced Engineering Mathematics (Wiley); R.K. Jain & S.R.K. Iyengar: Advanced Engineering Mathematics (Narosa).

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Contents

1. Recall & Foundations (10+2 Prerequisites)
2. Introduction to Partial Differentiation
3. Higher-Order Partial Derivatives & PDEs of Physics
4. Homogeneous Functions & The 3-Step Test
5. Euler's Theorem for Homogeneous Functions
6. Composite Functions & The Multivariable Chain Rule
7. Composite Euler's Theorem (When $F(u)$ is Homogeneous)
8. Total Differentiation & Tangent Plane Approximations
9. Implicit Function Differentiation
10. Jacobians & Coordinate Transformations
11. Properties of Jacobians
12. Taylor's & Maclaurin's Theorems for Two Variables
13. Standard Practice Problem Bank (15 Exam Questions)
14. Quick Revision Formula Reference & Exam Pitfall Guide

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1. Recall & Foundations (10+2 Prerequisites)

Partial differentiation is a direct multivariable generalization of single-variable calculus from school mathematics. Every classical rule—the power rule, product rule, quotient rule, and chain rule—remains valid, with one fundamental operating principle: differentiate with respect to one active variable while treating all other variables as strict constants.

1.1 Types of Functions

                                      FUNCTIONS
                                          |
                +-------------------------+-------------------------+
                |                                                   |
        EXPLICIT FUNCTIONS                                  IMPLICIT FUNCTIONS
         y = f(x) or z = f(x,y)                                 F(x,y,z) = 0
    (Dependent variable isolated)                      (Variables intertwined)

1. Algebraic Functions: Constructed from variables with constant rational exponents via addition, subtraction, multiplication, and division.
   Examples: $f(x) = x^3 - 4x + 7$, $\sqrt{x} = x^{1/2}$, $\dfrac{1}{x^2} = x^{-2}$.
2. Trigonometric Functions: Periodic circular ratios $\sin x, \cos x, \tan x, \cot x, \sec x, \csc x$ and their inverse relations $\sin^{-1} x, \cos^{-1} x, \tan^{-1} x$.
3. Exponential & Logarithmic Functions: Functions of the form $e^x$, $a^x$ ($a > 0, a \neq 1$), and the natural logarithm $\ln x = \log_e x$.
   Core Inversion Identities: $\ln(e^x) = x$ and $e^{\ln x} = x$.
4. Explicit Functions: The dependent variable is completely isolated on one side: $y = f(x)$ or $z = f(x,y)$.
5. Implicit Functions: The variables are intertwined within an equation: $F(x,y) = 0$ (e.g., $x^3 + y^3 - 3axy = 0$). Isolating one variable in closed algebraic form is difficult or impossible.
6. Composite Functions: A nested function of another function: $u = f(g(x))$ or $u = f(x(t), y(t))$. Differentiated using the Chain Rule.

1.2 Standard Single-Variable Differentiation Formulae

Function $f(x)$	Derivative $f'(x) = \dfrac{d}{dx}[f(x)]$	Function $f(x)$	Derivative $f'(x) = \dfrac{d}{dx}[f(x)]$
$x^n$ ($n \in \mathbb{R}$)	$n x^{n-1}$	$\sin x$	$\cos x$
$e^x$	$e^x$	$\cos x$	$-\sin x$
$a^x$ ($a > 0$)	$a^x \ln a$	$\tan x$	$\sec^2 x$
$\ln x$ ($x > 0$)	$\dfrac{1}{x}$	$\cot x$	$-\csc^2 x$
$\sin^{-1} x$ ($\vert x\vert < 1$)	$\dfrac{1}{\sqrt{1 - x^2}}$	$\sec x$	$\sec x \tan x$
$\cos^{-1} x$ ($\vert x\vert < 1$)	$-\dfrac{1}{\sqrt{1 - x^2}}$	$\csc x$	$-\csc x \cot x$
$\tan^{-1} x$	$\dfrac{1}{1 + x^2}$	$\cot^{-1} x$	$-\dfrac{1}{1 + x^2}$

1.3 Core Rules of Differentiation

1. Linearity: Constants factor out and derivatives distribute across sums:

2. Product Rule:

3. Quotient Rule:

4. Single-Variable Chain Rule: If $y = f(u)$ and $u = g(x)$, then:

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2. Introduction to Partial Differentiation

2.1 The Concept of a Partial Derivative

In single-variable calculus, the derivative $\dfrac{dy}{dx}$ measures the instantaneous rate of change of $y = f(x)$ along a 2D planar curve.

When a quantity depends on two or more independent variables (e.g., the volume of a cylinder $V = \pi r^2 h$, or temperature $T = T(x,y,z)$), we determine how the function changes when one variable varies while all other variables remain held strictly constant.

2.2 Formal Limit Definitions

Let $z = f(x,y)$ be defined on an open neighborhood in $\mathbb{R}^2$:

2.3 Geometric Intuition: Slopes of Tangents on a 3D Surface

The graph of $z = f(x,y)$ represents a two-dimensional surface suspended in 3D Cartesian space $\mathbb{R}^3$:

• Intersecting the surface $z = f(x,y)$ with the vertical plane $y = y_0$ produces a 2D trace curve $z = f(x, y_0)$. The partial derivative $\dfrac{\partial z}{\partial x}(x_0, y_0)$ is the slope of the tangent line to this trace curve in the positive $x$-direction.
• Similarly, intersecting with the vertical plane $x = x_0$ produces the trace curve $z = f(x_0, y)$. The partial derivative $\dfrac{\partial z}{\partial y}(x_0, y_0)$ is the slope of the tangent line in the positive $y$-direction.

Together, these two tangent vectors span the tangent plane to the surface at $(x_0, y_0, z_0)$.

2.4 Subscript Notation Guide

Notation	Leibniz Form	Verbal Pronunciation	Operational Procedure
$f_x$	$\dfrac{\partial f}{\partial x}$	"$f$ sub $x$"	Differentiate $f$ w.r.t. $x$; treat $y$ as a constant
$f_y$	$\dfrac{\partial f}{\partial y}$	"$f$ sub $y$"	Differentiate $f$ w.r.t. $y$; treat $x$ as a constant
$f_{xx}$	$\dfrac{\partial^2 f}{\partial x^2}$	"$f$ sub $xx$"	Differentiate $f_x$ again w.r.t. $x$
$f_{yy}$	$\dfrac{\partial^2 f}{\partial y^2}$	"$f$ sub $yy$"	Differentiate $f_y$ again w.r.t. $y$
$f_{xy}$	$\dfrac{\partial^2 f}{\partial y\,\partial x}$	"$f$ sub $xy$"	Take $f_x = \dfrac{\partial f}{\partial x}$, then differentiate w.r.t. $y$
$f_{yx}$	$\dfrac{\partial^2 f}{\partial x\,\partial y}$	"$f$ sub $yx$"	Take $f_y = \dfrac{\partial f}{\partial y}$, then differentiate w.r.t. $x$
$f_{xxy}$	$\dfrac{\partial^3 f}{\partial y\,\partial x^2}$	"$f$ sub $xxy$"	Differentiate $f_{xx}$ w.r.t. $y$

2.5 Clairaut's Theorem (Schwarz's Theorem)

Theorem: If the mixed second-order partial derivatives $f_{xy}$ and $f_{yx}$ of a function $f(x,y)$ are continuous on an open disc containing $(x_0, y_0)$, then:

Significance for engineering examinations: For all continuous elementary functions (polynomials, exponentials, sines, cosines), the order in which mixed partial derivatives are evaluated does not matter.

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Worked Examples — Basic Partial Differentiation

Example 2.1

Problem: Given $z = x^2 \cos y$, determine $\dfrac{\partial z}{\partial x}$, $\dfrac{\partial^2 z}{\partial x^2}$, $\dfrac{\partial z}{\partial y}$, and $\dfrac{\partial^2 z}{\partial x\,\partial y}$.

Solution:

1. To compute $\dfrac{\partial z}{\partial x}$, hold $y$ constant ($\cos y$ behaves as a scalar coefficient):

2. Differentiate $\dfrac{\partial z}{\partial x}$ again w.r.t. $x$:

3. To compute $\dfrac{\partial z}{\partial y}$, hold $x^2$ constant:

4. Compute the mixed partial $f_{xy} = \dfrac{\partial}{\partial y}\left(\dfrac{\partial z}{\partial x}\right)$:

Verification via $f_{yx}$: $\dfrac{\partial}{\partial x}(-x^2 \sin y) = -2x \sin y$. Matches $f_{xy}$ by Clairaut's theorem. $\blacksquare$

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Example 2.2

Problem: Given $z = \ln(\sin(x^2 y^3))$, determine $\dfrac{\partial z}{\partial x}$.

Solution: This expression contains three nested layers. Apply the multivariable chain rule from the outside inward:

1. Outer layer ($\ln u$): $\dfrac{\partial z}{\partial x} = \dfrac{1}{\sin(x^2 y^3)} \cdot \dfrac{\partial}{\partial x}[\sin(x^2 y^3)]$.
2. Middle layer ($\sin v$): $\dfrac{\partial}{\partial x}[\sin(x^2 y^3)] = \cos(x^2 y^3) \cdot \dfrac{\partial}{\partial x}[x^2 y^3]$.
3. Inner layer ($x^2 y^3$, holding $y^3$ constant): $\dfrac{\partial}{\partial x}[x^2 y^3] = 2x y^3$.
4. Multiply terms together and simplify using $\dfrac{\cos \theta}{\sin \theta} = \cot \theta$:

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3. Higher-Order Partial Derivatives & PDEs of Physics

3.1 Order and Degree of a Partial Differential Equation

• Order: The highest derivative order appearing anywhere within the differential equation.
• Degree: The algebraic power (exponent) to which the highest-order derivative is raised, after the equation has been cleared of fractional roots and radicals involving derivatives.

PDE Expression	Order	Degree	Technical Justification
$\dfrac{\partial^2 z}{\partial x^2} + \dfrac{\partial z}{\partial y} = 0$	2	1	Highest derivative is 2nd order, raised to power 1.
$\left(\dfrac{\partial^3 z}{\partial x^2\,\partial y}\right)^2 + z = 0$	3	2	Highest derivative is 3rd order, raised to power 2.
$\left(\dfrac{\partial z}{\partial x}\right)^4 + \dfrac{\partial^2 z}{\partial y^2} = 5$	2	1	Highest derivative is $\dfrac{\partial^2 z}{\partial y^2}$ (order 2), raised to power 1.

Critical Exam Distinction:

• $\dfrac{\partial^2 z}{\partial x^2}$ represents successive differentiation twice w.r.t. $x$ (curvature).
• $\left(\dfrac{\partial z}{\partial x}\right)^2$ represents taking the first derivative and squaring its numeric value.

3.2 The Three Classical Second-Order Linear PDEs of Physics

                                  CLASSICAL PDEs
                                        |
       +--------------------------------+--------------------------------+
       |                                |                                |
   LAPLACE EQUATION               HEAT EQUATION                    WAVE EQUATION
  d²u/dx² + d²u/dy² = 0            du/dt = c² d²u/dx²              d²u/dt² = c² d²u/dx²
 (Elliptic · Steady-State)      (Parabolic · Diffusion)          (Hyperbolic · Vibration)

1. Laplace's Equation (Elliptic):

• Governs steady-state temperature distributions, gravitational potential fields, and electrostatics.
• Solutions to Laplace's equation with continuous second derivatives are called harmonic functions.

2. Heat / Diffusion Equation (Parabolic):

• Governs the non-steady conduction of heat along a thermal rod, where $c^2$ is the thermal diffusivity coefficient.

3. Wave Equation (Hyperbolic):

• Governs mechanical vibrations, musical acoustic strings, and electromagnetic radiation, where $c$ represents the wave propagation speed.

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4. Homogeneous Functions & The 3-Step Test

4.1 Formal Mathematical Definition

A function $f(x,y)$ is defined as homogeneous of degree $n$ if scaling all independent variables by an arbitrary positive scalar factor $t > 0$ scales the output by $t^n$:

Equivalently, by choosing the scaling factor $t = \dfrac{1}{x}$, any two-variable homogeneous function can be expressed in canonical ratio form:

4.2 The 3-Step Test for Homogeneity

To verify if an expression is homogeneous and rapidly determine its degree $n$:

1. Step 1: Factor out the highest power of $x$ from the numerator and denominator independently.
2. Step 2: Combine all remaining terms inside parentheses. Every term must collapse exclusively into powers of the dimensionless ratio $(y/x)$. If any orphan term or disparate power ratio appears, the expression is non-homogeneous.
3. Step 3: Subtract the denominator power of $x$ from the numerator power. The resulting exponent is the degree $n$.

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Worked Examples — Testing Homogeneity & Finding Degree

Example 4.1

Expression: $f(x,y) = \dfrac{x^2 + y^2}{\sqrt{x} + \sqrt{y}}$

Test: Factor $x^2$ from the numerator and $x^{1/2}$ from the denominator:

Conclusion: Homogeneous of degree $n = \dfrac{3}{2}$. $\blacksquare$

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Example 4.2

Expression: $f(x,y) = \dfrac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}}$

Test: Factor $x^{1/3}$ from the numerator and $x^{1/2}$ from the denominator:

Conclusion: Homogeneous of degree $n = -\dfrac{1}{6}$. $\blacksquare$

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Example 4.3 (Non-Homogeneous Case)

Expression: $f(x,y) = \dfrac{x^2 + y^3}{\sqrt{x} + \sqrt{y}}$

Test: Factoring $x^2$ from the numerator yields $x^2\left[1 + \dfrac{y^3}{x^2}\right]$.
The ratio $\dfrac{y^3}{x^2} = x\left(\dfrac{y}{x}\right)^3$ cannot be written purely as a function of $(y/x)$ because an extra factor of $x$ remains.
Conclusion: Not Homogeneous. Euler's theorem cannot be applied. $\blacksquare$

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Example 4.4

Expression: $f(x,y) = \left(\dfrac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}}\right)^{1/2}$

Test: From Example 4.2, the inner bracket is $x^{-1/6} \phi(y/x)$. Raising to power $1/2$:

Conclusion: Homogeneous of degree $n = -\dfrac{1}{12}$. $\blacksquare$

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Example 4.5

Expression: $u = \dfrac{x^2 y^2}{x^{1/3} + y^{1/3}}$

Test: Write numerator as $x^4\left(\dfrac{y}{x}\right)^2$ and denominator as $x^{1/3}\left[1 + \left(\dfrac{y}{x}\right)^{1/3}\right]$:

Conclusion: Homogeneous of degree $n = \dfrac{11}{3}$. $\blacksquare$

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5. Euler's Theorem for Homogeneous Functions

5.1 First-Order Euler Identity

Euler's Theorem: If $u = f(x,y)$ is a homogeneous function of degree $n$ possessing continuous first-order partial derivatives, then:

5.2 Step-by-Step Formal Proof of First-Order Euler

Since $u$ is homogeneous of degree $n$, we write:

Step 1: Differentiate $u$ partially with respect to $x$ using the product and chain rules:

Since $\dfrac{\partial t}{\partial x} = -\dfrac{y}{x^2}$:

Step 2: Multiply both sides by $x$:

Step 3: Differentiate $u$ partially with respect to $y$:

Since $\dfrac{\partial t}{\partial y} = \dfrac{1}{x}$:

Step 4: Multiply both sides by $y$:

Step 5: Add Equation 1 and Equation 2:

The opposing $\phi'(t)$ terms cancel out completely:

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5.3 Second-Order Euler Identity

Second-Order Euler Identity:

Proof:

Differentiate the first-order identity $x u_x + y u_y = n u$ with respect to $x$:

Multiply by $x$:

Now differentiate $x u_x + y u_y = n u$ with respect to $y$:

Multiply by $y$:

Add (A) and (B), applying Clairaut's theorem ($u_{yx} = u_{xy}$):

Since $x u_x + y u_y = n u$:

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Worked Examples — Direct Euler's Theorem

Example 5.1

Problem: Find $x \dfrac{\partial u}{\partial x} + y \dfrac{\partial u}{\partial y}$ where $u = \dfrac{x^3 + y^3}{\sqrt{x} + \sqrt{y}}$.

Solution:

1. Check homogeneity: $u = \dfrac{x^3[1 + (y/x)^3]}{x^{1/2}[1 + (y/x)^{1/2}]} = x^{5/2} \phi(y/x)$. Degree $n = \dfrac{5}{2}$.
2. Apply Euler's Theorem directly:

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Example 5.2

Problem: Find $x \dfrac{\partial u}{\partial x} + y \dfrac{\partial u}{\partial y}$ where $u = \dfrac{x^2 y^2}{x^{1/3} + y^{1/3}}$.

Solution: From Example 4.5, $n = \dfrac{11}{3}$.
Applying Euler: $x u_x + y u_y = \dfrac{11}{3} u$. $\blacksquare$

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6. Composite Functions & The Multivariable Chain Rule

A composite function occurs when a dependent variable $u$ is expressed in terms of intermediate variables $x, y, z$, which themselves depend on one or more independent parameters $t, r, s$.

         ONE PARAMETER                             TWO PARAMETERS
               u                                         u
             /   \                                     /   \
            x     y                                   x     y
             \   /                                   / \   / \
               t                                    r   s r   s
    du/dt = u_x x_t + u_y y_t             u_r = u_x x_r + u_y y_r
                                          u_s = u_x x_s + u_y y_s

6.1 Formulae for Multivariable Chain Rules

1. One Independent Parameter ($x = x(t), y = y(t)$):

2. Two Independent Parameters ($x = x(r,s), y = y(r,s)$):

3. Three Intermediate Variables ($u = f(x,y,z)$ with parameters $r, t$):

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Worked Examples — Multivariable Chain Rule

Example 6.1

Problem: Find $\dfrac{du}{dt}$ given $u = 3x^3 + 4x^2 - 6x$, where $x = t^2 + 1$.

Solution:

1. Compute partials: $\dfrac{du}{dx} = 9x^2 + 8x - 6$, and $\dfrac{dx}{dt} = 2t$.
2. Apply chain rule:

3. Expand:

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Example 6.2

Problem: If $u = x^2 + y^2 - z^2$, with $x = r + t$, $y = r - t$, and $z = 2rt$, find $\dfrac{\partial u}{\partial r}$, $\dfrac{\partial u}{\partial t}$, and $\dfrac{\partial^2 u}{\partial r^2}$.

Solution:

1. Compute all intermediate derivatives:
   • $u_x = 2x$, $u_y = 2y$, $u_z = -2z$.
   • $x_r = 1$, $y_r = 1$, $z_r = 2t$.
   • $x_t = 1$, $y_t = -1$, $z_t = 2r$.
2. Compute $\dfrac{\partial u}{\partial r}$:

3. Compute $\dfrac{\partial u}{\partial t}$:

4. Compute second-order partial $\dfrac{\partial^2 u}{\partial r^2}$:

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Example 6.3 (Famous University Proof)

Problem: If $u = f(s - t, \; t - r, \; r - s)$, prove that:

Proof: Let the intermediate arguments be $p = s - t$, $q = t - r$, and $w = r - s$.
Then $u = f(p, q, w)$. Apply the multivariable chain rule for each parameter:

1. Differentiate w.r.t. $r$:

2. Differentiate w.r.t. $s$:

3. Differentiate w.r.t. $t$:

Summing the three equations:

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7. Composite Euler's Theorem (When $F(u)$ is Homogeneous)

7.1 Mathematical Foundation

Frequently in university examinations, a given function $u$ is not homogeneous, but applying a function $F(u)$ produces a homogeneous function $z = F(u)$ of degree $n$ (common examples include $u = \sin^{-1}(\dots), \cos^{-1}(\dots), \tan^{-1}(\dots), \ln(\dots)$).

Since $z = F(u)$ is homogeneous of degree $n$, Euler's theorem applies to $z$:

Using the chain rule: $z_x = F'(u) u_x$ and $z_y = F'(u) u_y$. Substituting:

Dividing both sides by $F'(u)$:

7.2 The Composite Euler Formulas

1. First-Order Composite Identity ($uE'$):

2. Second-Order Composite Identity ($uE^2$):

where $g'(u) = \dfrac{dg}{du}$.

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Worked Examples — Composite Euler

Example 7.1

Problem: Given $u = \sin^{-1}\left(\dfrac{x^2 + y^2}{x + y}\right)$, find $x u_x + y u_y$ and $x^2 u_{xx} + 2xy u_{xy} + y^2 u_{yy}$.

Solution:

1. Isolate the homogeneous component:

Here $F(u) = \sin u$, and degree $n = 1$. 2. Calculate $F'(u) = \cos u$:

3. First-order result ($uE'$):

4. Compute $g'(u) = \dfrac{d}{du}(\tan u) = \sec^2 u$.
5. Second-order result ($uE^2$):

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Example 7.2

Problem: Given $u = \cos^{-1}\left(\dfrac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}}\right)$, evaluate $x u_x + y u_y$ and $x^2 u_{xx} + 2xy u_{xy} + y^2 u_{yy}$.

Solution:

1. Let $z = \cos u = \dfrac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}}$. From Example 4.2, degree $n = -\dfrac{1}{6}$.
2. Here $F(u) = \cos u$, and $F'(u) = -\sin u$:

3. First-order result:

4. Differentiate $g(u)$:

5. Second-order result:

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8. Total Differentiation & Tangent Plane Approximations

8.1 Total Differential

For a function $u = f(x,y)$, the total differential $du$ represents the principal linear change in $u$ caused by infinitesimal increments $dx$ and $dy$:

For three variables $u = f(x,y,z)$:

8.2 Geometric Significance: Tangent Plane Equation

The total differential $dz = f_x dx + f_y dy$ provides the linear approximation to the surface $z = f(x,y)$.
At a point $(x_0, y_0, z_0)$ on the surface, setting $\Delta z = z - z_0$, $dx = x - x_0$, and $dy = y - y_0$ yields the equation of the tangent plane:

8.3 Engineering Error Propagation

If independent variables $x$ and $y$ are measured with small experimental uncertainties $\delta x$ and $\delta y$:

• Absolute Error: $\delta u \approx du = \dfrac{\partial u}{\partial x}\delta x + \dfrac{\partial u}{\partial y}\delta y$
• Relative Error: $\dfrac{\delta u}{u} \approx \dfrac{du}{u} = \dfrac{1}{u}\left[\dfrac{\partial u}{\partial x}\delta x + \dfrac{\partial u}{\partial y}\delta y\right]$
• Percentage Error: $\dfrac{\delta u}{u} \times 100\%$

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Worked Examples — Total Differentiation

Example 8.1

Problem: Find $du$ where $u = x^2 + y^2 + 2xy$.

Solution:

1. Compute partials: $u_x = 2x + 2y$, and $u_y = 2y + 2x$.
2. Assemble total differential:

Verification: $u = (x + y)^2 \implies du = 2(x + y) d(x + y)$. $\blacksquare$

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9. Implicit Function Differentiation

An implicit function is defined by $f(x,y) = 0$, where $y$ is implicitly defined as a function of $x$.

9.1 First Derivative Formula

Taking the total differential of $f(x,y) = 0$:

Dividing across by $dx$ (assuming $\dfrac{\partial f}{\partial y} \neq 0$):

9.2 Second Derivative Formula

Differentiating $\dfrac{dy}{dx} = -\dfrac{f_x}{f_y}$ w.r.t. $x$ using the quotient rule:

Using total derivatives $\dfrac{d}{dx}(f_x) = f_{xx} + f_{xy}\dfrac{dy}{dx} = f_{xx} - f_{xy}\dfrac{f_x}{f_y}$ and $\dfrac{d}{dx}(f_y) = f_{yx} + f_{yy}\dfrac{dy}{dx} = f_{xy} - f_{yy}\dfrac{f_x}{f_y}$:

Exam Caution: Note the leading negative sign on the entire fraction.

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Worked Examples — Implicit Functions

Example 9.1 (Folium of Descartes)

Problem: Find $\dfrac{dy}{dx}$ for $x^3 + y^3 - 3axy = 0$.

Solution:

1. Let $f(x,y) = x^3 + y^3 - 3axy = 0$.
2. Compute partials:

3. Apply implicit formula:

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10. Jacobians & Coordinate Transformations

10.1 Formal Definition

The Jacobian of two functions $u(x,y)$ and $v(x,y)$ with respect to $x$ and $y$ is the determinant of the matrix of their first-order partial derivatives:

For three functions $u, v, w$ in terms of $x, y, z$:

10.2 Geometric Significance: Area and Volume Scaling

In multivariable integration and coordinate transformations, the Jacobian represents the local area or volume magnification factor:

• In 2D plane transformations: $dx\,dy = |J| \, du\,dv$.
• In 3D space transformations: $dx\,dy\,dz = |J| \, du\,dv\,dw$.

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Worked Examples — Jacobians

Example 10.1 (Polar Coordinate Transformation)

Problem: For $x = r \cos\theta, y = r \sin\theta$, calculate $J = \dfrac{\partial(x,y)}{\partial(r,\theta)}$.

Solution:

Application: This is the exact mathematical reason why $dx\,dy$ transforms into $r\,dr\,d\theta$ in polar integration. $\blacksquare$

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Example 10.2

Problem: Given $u = \dfrac{yz}{x}$, $v = \dfrac{xz}{y}$, $w = \dfrac{xy}{z}$, evaluate $\dfrac{\partial(u,v,w)}{\partial(x,y,z)}$.

Solution: Compute all 9 partial derivatives:

Evaluate the determinant:

Expanding along Row 1:

• Cofactor of $u_x$: $\left(-\frac{xz}{y^2}\right)\left(-\frac{xy}{z^2}\right) - \left(\frac{x}{y}\right)\left(\frac{x}{z}\right) = \frac{x^2}{yz} - \frac{x^2}{yz} = 0$.
• Cofactor of $u_y$: $\frac{z}{y}\left(-\frac{xy}{z^2}\right) - \frac{x}{y}\left(\frac{y}{z}\right) = -\frac{x}{z} - \frac{x}{z} = -\frac{2x}{z}$.
• Cofactor of $u_z$: $\frac{z}{y}\left(\frac{x}{z}\right) - \left(-\frac{xz}{y^2}\right)\left(\frac{y}{z}\right) = \frac{x}{y} + \frac{x}{y} = \frac{2x}{y}$.

Combine:

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11. Properties of Jacobians

11.1 Property 1: Chain Rule for Jacobians

If $u, v$ are functions of $x, y$, and $x, y$ are functions of $r, s$:

11.2 Property 2: Reciprocal / Inverse Property

If $u, v$ are functions of $x, y$, then:

Strategic Exam Benefit: If calculating $\dfrac{\partial(u,v)}{\partial(x,y)}$ directly involves intractable algebra, calculate $J' = \dfrac{\partial(x,y)}{\partial(u,v)}$ and simply invert it: $J = \dfrac{1}{J'}$.

11.3 Property 3: Functional Dependence

A set of functions $u(x,y)$ and $v(x,y)$ are functionally dependent (meaning an algebraic relation $\Phi(u,v) = 0$ connects them) if and only if their Jacobian vanishes identically:

11.4 Property 4: Triangular Jacobian Systems

If $u = f(x)$, $v = g(x,y)$, and $w = h(x,y,z)$, the partial derivative matrix is lower-triangular, so its determinant is simply the product of its diagonal entries:

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12. Taylor's & Maclaurin's Theorems for Two Variables

12.1 Standard Single-Variable Maclaurin Series (Prerequisites)

• $e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dots$ ($\forall x \in \mathbb{R}$)
• $\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dots$ (Odd powers)
• $\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dots$ (Even powers)
• $\ln(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dots$ ($-1 < x \le 1$)
• $(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!} x^2 + \dots$ ($|x| < 1$)

12.2 Two-Variable Taylor Series Theorem

Let $f(x,y)$ have continuous partial derivatives up to order $n$ in a region around $(a,b)$. The Taylor series of $f(x,y)$ in powers of $(x - a)$ and $(y - b)$ (where $h = x - a$ and $k = y - b$) is:

12.3 Two-Variable Maclaurin Series

When expanding about the origin $(a, b) = (0, 0)$, $h = x$ and $k = y$:

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Worked Examples — Series Expansions

Example 12.1

Problem: Expand $f(x,y) = e^x \sin y$ in a Maclaurin series about $(0,0)$ up to third-order terms.

Solution: Construct the partial derivative evaluation table at $(0,0)$:

Order	Derivative	Analytical Expression	Value at $(0,0)$
0	$f$	$e^x \sin y$	0
1	$f_x$	$e^x \sin y$	0
1	$f_y$	$e^x \cos y$	1
2	$f_{xx}$	$e^x \sin y$	0
2	$f_{yy}$	$-e^x \sin y$	0
2	$f_{xy}$	$e^x \cos y$	1
3	$f_{xxx}$	$e^x \sin y$	0
3	$f_{yyy}$	$-e^x \cos y$	-1
3	$f_{xxy}$	$e^x \cos y$	1
3	$f_{xyy}$	$-e^x \sin y$	0

Assemble the terms:

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Example 12.2

Problem: Expand $f(x,y) = x^2 e^y$ in powers of $(x - 1)$ and $(y - 2)$ up to second-degree terms (Taylor expansion about $(1,2)$).

Solution: Here $a = 1, b = 2$, with increments $h = x - 1$ and $k = y - 2$.

Derivative	Expression	Value at $(1,2)$
$f$	$x^2 e^y$	$1^2 e^2 = \mathbf{e^2}$
$f_x$	$2x e^y$	$2(1)e^2 = \mathbf{2e^2}$
$f_y$	$x^2 e^y$	$1^2 e^2 = \mathbf{e^2}$
$f_{xx}$	$2e^y$	$\mathbf{2e^2}$
$f_{yy}$	$x^2 e^y$	$\mathbf{e^2}$
$f_{xy}$	$2x e^y$	$\mathbf{2e^2}$

Assemble:

Exam Tip: Retain powers of $(x - 1)$ and $(y - 2)$ intact. Do not expand or substitute numeric values for $h$ and $k$.

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13. Standard Practice Problem Bank (15 Exam Questions)

Group A: Partial Derivatives — Direct Computation

P1

Problem: If $u = x^3 + y^3 + z^3 - 3xyz$, show that $\dfrac{\partial u}{\partial x} + \dfrac{\partial u}{\partial y} + \dfrac{\partial u}{\partial z} = 3(x + y + z)^2$.
Hint: Compute $u_x = 3x^2 - 3yz$, $u_y = 3y^2 - 3xz$, $u_z = 3z^2 - 3xy$. Sum them and factor using the identity $(x+y+z)^2 = x^2+y^2+z^2 + 2(xy+yz+zx)$.

P2

Problem: If $z(x+y) = x^2 + y^2$, show that:

Hint: Write $z = \dfrac{x^2+y^2}{x+y}$. Compute $z_x$ and $z_y$ via quotient rule, then evaluate both sides algebraically.

P3

Problem: If $u = \log(\tan x + \tan y + \tan z)$, prove that:

Hint: Notice $\sin 2x \cdot u_x = \sin 2x \cdot \dfrac{\sec^2 x}{\sum \tan x} = \dfrac{2\sin x \cos x / \cos^2 x}{\sum \tan x} = \dfrac{2\tan x}{\sum \tan x}$. Sum across $x,y,z$.

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Group B: Euler's Theorem & Applications

P4

Problem: If $u = \tan^{-1}\left(\dfrac{x^3 + y^3}{x - y}\right)$, prove that $x \dfrac{\partial u}{\partial x} + y \dfrac{\partial u}{\partial y} = \sin 2u$.
Hint: $\tan u = \dfrac{x^3+y^3}{x-y}$ is homogeneous of degree $n = 2$. Apply Composite Euler: $g(u) = 2\dfrac{\tan u}{\sec^2 u} = 2\sin u\cos u = \sin 2u$.

P5

Problem: If $u = \sin^{-1}\left(\dfrac{x^2 + y^2}{x + y}\right)$, find $x^2 u_{xx} + 2xy u_{xy} + y^2 u_{yy}$.
Answer: Degree $n = 1$, $g(u) = \tan u$, $g'(u) = \sec^2 u$. Result is $g(u)[g'(u)-1] = \tan u(\sec^2 u - 1) = \tan^3 u$.

P6

Problem: Verify Euler's theorem for $f(x,y) = \sqrt{x^2 + xy + y^2}$.
Hint: Factor $x$ to show $n = 1$. Calculate $f_x$ and $f_y$ directly, multiply by $x$ and $y$, and verify $x f_x + y f_y = f$.

P7

Problem: If $u = \log\left(\dfrac{x^2 + y^2}{x + y}\right)$, prove $x u_x + y u_y = 1$.
Hint: $e^u = \dfrac{x^2+y^2}{x+y}$ is homogeneous of degree 1. $g(u) = 1 \cdot \dfrac{e^u}{e^u} = 1$.

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Group C: Chain Rule & Composite Functions

P8

Problem: If $z = f(x,y)$, where $x = e^u + e^{-v}$ and $y = e^{-u} + e^v$, prove:

Hint: Apply chain rule: $z_u = z_x(e^u) + z_y(-e^{-u})$, and $z_v = z_x(-e^{-v}) + z_y(e^v)$. Subtract.

P9

Problem: If $r^2 = x^2 + y^2 + z^2$, evaluate $\dfrac{\partial r}{\partial x}$, $\dfrac{\partial r}{\partial y}$, and $\dfrac{\partial r}{\partial z}$.
Answer: Differentiating implicitly w.r.t. $x$: $2r r_x = 2x \implies r_x = \dfrac{x}{r}$. Similarly, $r_y = \dfrac{y}{r}$, $r_z = \dfrac{z}{r}$.

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Group D: Jacobians & Transformations

P10

Problem: If $u = x^2 - 2y$ and $v = x + y$, find $J = \dfrac{\partial(u,v)}{\partial(x,y)}$ and verify the inverse property.
Answer: $J_1 = \begin{vmatrix} 2x & -2 \\ 1 & 1 \end{vmatrix} = 2x + 2 = 2(x+1)$. Thus $J_2 = \dfrac{1}{2(x+1)}$.

P11

Problem: If $u + v = x^2$ and $uv = y$, find $\dfrac{\partial(u,v)}{\partial(x,y)}$ using the reciprocal property.
Hint: Differentiate w.r.t. $u,v$ to find $\dfrac{\partial(x,y)}{\partial(u,v)}$, then take the reciprocal.

P12

Problem: Are the elementary symmetric polynomials $u = x + y + z$, $v = xy + yz + zx$, and $w = xyz$ functionally dependent?
Answer: Compute $J = \dfrac{\partial(u,v,w)}{\partial(x,y,z)} = (x - y)(y - z)(z - x) \neq 0$. Therefore, they are functionally independent.

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Group E: Taylor's & Maclaurin's Series

P13

Problem: Expand $e^{x+y}$ in a Maclaurin series about $(0,0)$ up to 2nd-order terms.
Answer: All partials at $(0,0)$ equal 1. $e^{x+y} = 1 + (x + y) + \dfrac{1}{2}(x^2 + 2xy + y^2) + \dots$

P14

Problem: Expand $f(x,y) = \sin(xy)$ about $(0,0)$ up to 3rd-order terms.
Answer: All derivatives vanish at the origin except $f_{xy}(0,0) = 1$. The expansion is simply $f(x,y) = xy + \dots$

P15

Problem: Expand $f(x,y) = \tan^{-1}(y/x)$ in powers of $(x - 1)$ and $(y - 1)$ up to second degree.
Hint: At $(1,1)$, $f(1,1) = \pi/4$, $f_x = -1/2$, $f_y = 1/2$, $f_{xx} = 1/2$, $f_{yy} = -1/2$, $f_{xy} = 0$.

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14. Quick Revision Formula Reference & Exam Pitfall Guide

14.1 Master Formula Reference Table

Core Calculus Topic	Mathematical Formula	Key Operational Condition
Partial Derivative	$\dfrac{\partial f}{\partial x} = \lim_{h \to 0} \dfrac{f(x+h, y) - f(x,y)}{h}$	Treat $y$ as a strict constant
Clairaut's Theorem	$f_{xy} = f_{yx}$	Continuous second partial derivatives
Homogeneous Function	$f(tx, ty) = t^n f(x,y) \iff f = x^n \phi(y/x)$	Degree $n \in \mathbb{R}$
Euler 1st Order	$x u_x + y u_y = n u$	$u$ is homogeneous of degree $n$
Euler 2nd Order	$x^2 u_{xx} + 2xy u_{xy} + y^2 u_{yy} = n(n-1)u$	$u$ is homogeneous of degree $n$
Composite Euler ($uE'$)	$x u_x + y u_y = g(u) = n \dfrac{F(u)}{F'(u)}$	$F(u)$ is homogeneous of degree $n$
Composite Euler ($uE^2$)	$x^2 u_{xx} + 2xy u_{xy} + y^2 u_{yy} = g(u)[g'(u) - 1]$	Same as above; $g'(u) = \dfrac{dg}{du}$
Total Differential	$du = \dfrac{\partial u}{\partial x} dx + \dfrac{\partial u}{\partial y} dy$	First-order differentials
Chain Rule (1 parameter)	$\dfrac{du}{dt} = \dfrac{\partial u}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial u}{\partial y}\dfrac{dy}{dt}$	$x = x(t), y = y(t)$
Implicit Derivative ($y'$)	$\dfrac{dy}{dx} = -\dfrac{f_x}{f_y}$	$f(x,y) = 0, \; f_y \neq 0$
Implicit Derivative ($y''$)	$\dfrac{d^2 y}{dx^2} = -\dfrac{f_{xx} f_y^2 - 2f_x f_y f_{xy} + f_{yy} f_x^2}{f_y^3}$	Note the leading minus sign
2D Jacobian	$J = \dfrac{\partial(u,v)}{\partial(x,y)} = \begin{vmatrix} u_x & u_y \\ v_x & v_y \end{vmatrix}$	Row = Function, Column = Variable
Jacobian Reciprocal	$\dfrac{\partial(u,v)}{\partial(x,y)} \cdot \dfrac{\partial(x,y)}{\partial(u,v)} = 1$	Invertible mappings
Functional Dependence	$\dfrac{\partial(u,v)}{\partial(x,y)} = 0 \iff$ dependent	Implies $\Phi(u,v) = 0$ exists
Polar Jacobian	$\dfrac{\partial(x,y)}{\partial(r,\theta)} = r$	$x = r\cos\theta, y = r\sin\theta$
Spherical Jacobian	$\dfrac{\partial(x,y,z)}{\partial(r,\theta,\phi)} = -r^2 \sin\phi$	Volume element $\vert J\vert = r^2\sin\phi$
2-Var Taylor Series	$f(a,b) + [h f_x + k f_y] + \frac{1}{2!}[h^2 f_{xx} + 2hk f_{xy} + k^2 f_{yy}] + \dots$	$h = x - a, \; k = y - b$
2-Var Maclaurin Series	$f(0,0) + [x f_x + y f_y] + \frac{1}{2!}[x^2 f_{xx} + 2xy f_{xy} + y^2 f_{yy}] + \dots$	Expanded about the origin

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14.2 Top 8 Critical Exam Pitfalls — Do Not Lose Marks Here

1. Failure to verify homogeneity: Never blindly apply Euler's theorem ($x u_x + y u_y = n u$) before verifying that all terms share the same degree. If terms have unequal powers (e.g., $x^2 + y^3$), the function is non-homogeneous and Euler's theorem does not hold.
2. Confusing $u$ with $F(u)$: In questions like $u = \tan^{-1}\left(\dfrac{x^3+y^3}{x-y}\right)$, $u$ is not homogeneous. You must isolate $F(u) = \tan u$, find its degree, and apply the composite identity $g(u) = n \dfrac{F(u)}{F'(u)}$.
3. Omitting the negative sign in implicit derivatives: Both $\dfrac{dy}{dx} = -\dfrac{f_x}{f_y}$ and $\dfrac{d^2 y}{dx^2} = -\dfrac{f_{xx}f_y^2 - 2f_x f_y f_{xy} + f_{yy}f_x^2}{f_y^3}$ carry an essential leading minus sign.
4. Substituting $h$ and $k$ in Taylor series: Always leave the answer in powers of $(x - a)$ and $(y - b)$. Never replace $h = x - a$ with a single constant number; doing so destroys the Taylor polynomial.
5. Transposing Jacobian matrices: In a Jacobian matrix, Row $i$ contains partial derivatives of the $i$-th function, and Column $j$ contains derivatives with respect to the $j$-th variable. Swapping rows and columns transposes the matrix, which reverses the sign of an odd-order permutation.
6. Confusing $\sin^{-1} x$ with $(\sin x)^{-1}$: $\sin^{-1} x = \arcsin x$ (inverse function), with derivative $\dfrac{1}{\sqrt{1-x^2}}$. $(\sin x)^{-1} = \csc x$ (reciprocal function), with derivative $-\csc x \cot x$.
7. Second partial vs. squared derivative: $\dfrac{\partial^2 z}{\partial x^2}$ is curvature (differentiating twice); $\left(\dfrac{\partial z}{\partial x}\right)^2$ is the square of the slope. They are fundamentally different operations.
8. Composite Euler second-order multiplier: Remember the factor is $g(u)[g'(u) - 1]$, not $g(u) g'(u)$. The $-1$ comes from bringing the first-order identity over during differentiation.