Engineering Mathematics Short Notes | Functions of Two Variables – Limits, Continuity, Partial Derivatives, Lagrange Multipliers, Double Integrals

ENGINEERING MATHEMATICS – FUNCTIONS OF TWO VARIABLES

(Complete Short Notes Limit, continuity and partial derivatives; Directional derivative; Total derivative; Maxima, minima and saddle points; Method of Lagrange multipliers; Double integrals and their applications for University Exams & GATE)

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1. INTRODUCTION

A function of two variables is a rule that assigns to each ordered pair 

$(x, y)$ in a domain $D \subset \mathbb{R}^2$ a real number $z$.

$$z = f(x, y)$$

Examples: $z = x^2 + y^2$, $z = e^{xy}$, $z = \sin(x + y)$

Such functions appear in Engineering fields like Heat Transfer, Fluid Mechanics, Optimization, Machine Learning, and Control Systems.

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2. LIMIT OF A FUNCTION OF TWO VARIABLES

2.1 Definition

A function 

$f(x, y)$ has limit $L$ at $(a, b)$if

$$\lim_{(x,y)\to(a,b)} f(x,y) = L$$

whenever $(x, y)$ approaches $(a, b)$ along any path.

2.2 Why limits of two variables are harder?

Because in 2D, infinitely many paths approach a point:

1. Straight line: $y=mx$

2. Parabola: $y = kx^2$
   

3. Curve: $x = t\cos t, y = t\sin t$ etc.

If limit differs along different paths, limit does not exist.

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2.3 Standard results

(i)

$$\underset{(x,y)\to (0,0)}{\lim }\frac{x^2+y^2}{x^2+y^2+1}=0$$

(ii) Limits typically fail if numerator/denominator mix non-homogeneous polynomials.

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2.4 Method of Checking Limit

1. Substitute 

straight-line path $y=mx$

1. Try parabolic path $y = kx^2$
   

2. Convert to polar form:

$$x=r\cos \theta ,y=r\sin \theta$$

If limit is independent of $\theta$, limit exists.

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2.5 Solved Example (GATE Type)

Example:
Evaluate

$$\underset{(x,y)\to (0,0)}{\lim }\frac{x^{2}y}{x^4+y^2}\mathrm{.}$$

Solution:
Take path $y = x^2$:

$$f(x, x^2) = \frac{x^2(x^2)}{x^4 + x^4} = \frac{x^4}{2x^4} = \frac12$$

Take path $y = 0$:

$$f(x,0)=0$$

Limits differ → limit does not exist.

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3. CONTINUITY OF TWO VARIABLE FUNCTIONS

Definition

$f(x, y)$ is continuous at $(a, b)$ if:

1. $f(a, b)$ exists

2. $\lim f(x, y)$ exists

3. Limit equals value

$$\underset{(x,y)\to (a,b)}{\lim }f(x,y)=f(a,b)$$

Discontinuity occurs when:

1. Undefined at point

2. Limit does not exist

3. Limit exists but differs from value

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4. PARTIAL DERIVATIVES

Definition

Partial derivative w.r.t. $x$:

$$\frac{\mathrm{\partial }f}{\mathrm{\partial }x}=\underset{h\to 0}{\lim }\frac{f(x+h,y)-f(x,y)}{h}$$

Partial derivative w.r.t. $y$:

$$\frac{\mathrm{\partial }f}{\mathrm{\partial }y}=\underset{k\to 0}{\lim }\frac{f(x,y+k)-f(x,y)}{k}$$

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4.1 Higher-order partials

$$f_{xx}=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2},f_{xy}=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y},f_{yx}=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }y\mathrm{\partial }\mathrm{x}}$$

Clairaut’s Theorem

If $f$ is continuous near $(a,b)$:

$$f_{xy} = f_{yx}$$

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4.2 Solved Example

Find partial derivatives of:

$$f(x,y)=x^2y+e^{xy}$$
$$f_x = 2xy + ye^{xy}$$
$$f_y=x^2+x{e}^{xy}$$

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5. DIRECTIONAL DERIVATIVE

Definition

Directional derivative in direction 

$\mathbf{u} = \langle u_1, u_2\rangle$ is:

$$D_{\mathbf{u}}f = f_x u_1 + f_y u_2$$

Direction vector must be unit:

$$\mathbf{u}=\frac{\mathbf{a}}{\mathrm{\parallel }\mathbf{a}\mathrm{\parallel }}$$

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Geometric Meaning

Directional derivative gives the rate of change of a surface z=f(x,y) in any direction.

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Solved Example

For 

$f(x,y)=x^2\mathrm{y,\; find\; directional\; derivative\; at }$$(1,2)$ in direction of $\mathbf{a} = \langle 3,4\rangle$.

1. Normalize:

$$\mathbf{u}=\frac{1}{5}(3,4)$$

2. Compute partials:

$$f_x = 2xy = 4$$
$$f_y=x^2=1$$

3. Directional derivative:

$$D_{\mathbf{u}}f = 4\cdot\frac35 + 1\cdot\frac45 = \frac{16}{5}$$

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6. TOTAL DERIVATIVE

If

$$z=f(x,y),x=x(t),y=y(t)$$

then

$$\frac{dz}{dt} = f_x\frac{dx}{dt} + f_y\frac{dy}{dt}$$

This is used in:

• Thermodynamics

• Control theory

• Transport equations

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7. EXTREMA OF FUNCTIONS OF TWO VARIABLES

(Local maxima, minima, saddle points)

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7.1 First-Order Condition

At extremum:

$$f_x=0,f_y=0$$

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7.2 Second-Order Condition

Compute:

$$D=f_{xx}{f}_{yy}-(f_{xy}{)}^2$$

Check:

Condition	Nature
(D > 0) and fxx > 0)	Local Minimum
(D > 0) and fxx < 0)	Local Maximum
(D < 0)	Saddle Point
(D = 0)	Test Fails

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7.3 Example (GATE Level)

Let

$$f(x,y)=x^2+xy+y^2$$

Compute partials:

$$f_x = 2x + y = 0$$
$$f_y=x+2y=0$$

Solving pair:

$$x=y=0$$

Second-order test:

$$f_{xx}=2,\quad f_{yy}=2,\quad f_{xy}=1$$
$$D=(2)(2)-1^2=3>0,f_{xx}>0$$

∴ Local Minimum at (0,0)

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8. LAGRANGE MULTIPLIERS

Used to find maxima/minima subject to constraint

$g(x,y)=0$.

Rule

Solve:

$$\nabla f = \lambda \nabla g$$
$$g(x,y)=0$$

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Example (University Level)

Maximize

$$f(x,y)=xy$$

subject to

$$x^2+y^2=1$$

1. Compute:

$$\nabla f = (y,x)$$
$$\mathrm{\nabla }g=(2x,2y)$$

2. Set:

$$(y,x)=\lambda (2x,2y)$$

Solve system →

$$x=\pm \frac{1}{\sqrt{2}},y=\pm \frac{1}{\sqrt{2}}$$

Maximum value:

$$f = \frac12$$

Minimum:

$$f=-\frac{1}{2}$$

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9. DOUBLE INTEGRALS

Used to compute:

1. Area

2. Volume under a surface

3. Mass (variable density)

4. Probability distributions

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9.1 Definition

$${\iint }_{R}f(x,y)dA$$

9.2 Iterated Form

$${\int }_a^b{\int }_{g_1(x)}^{g_2(x)}f(x,y)dydx$$

or

$$\int_{c}^{d}\int_{h_1(y)}^{h_2(y)} f(x,y)\,dx\,dy$$

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10. APPLICATIONS OF DOUBLE INTEGRALS

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10.1 Area of a Region

$$A={\iint }_{R}1dA$$

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10.2 Volume under a surface

$$V={\iint }_{R}f(x,y)dA$$

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10.3 Solved Example

Evaluate

$${\iint }_R(x+y)dA$$

over region $0\le x\le1$, $0\le y\le2$.

Solution:

$$\int_0^1\int_0^2 (x+y)\,dy\,dx$$

Inner integral:

$$\int_0^2 (x+y)\, dy = 2x + 2$$

Then:

$${\int }_0^1(2x+2)dx={[x^2+2x]}_0^1=1+2=3$$

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12. EXAM TIPS (UNIVERSITY + GATE)

High-Weightage Concepts

✔ Limits using polar substitution
✔ Directional derivative formula
✔ Lagrange multipliers
✔ Double integral region transformation
✔ Classification of critical points using Hessian
✔ Changing the order of integration

Common Mistakes

❌ Forgetting to normalize direction vectors
❌ Using wrong region limits in double integrals
❌ Incorrectly identifying saddle points
❌ Assuming limit exists without checking multiple paths

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13. SUPER QUICK SHORT NOTES (FOR LAST-DAY REVISION)

1. Limit exists ↔ Same value along all paths

2. Partial derivatives treat other variables as constant

3. Directional derivative = ∇f · unit vector

4. Extrema: solve $f_x=0, f_y=0$ → apply Hessian test

5. Lagrange multipliers: ∇f = λ∇g

6. Double integrals compute area/volume

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