These functions represent relationships where output depends on two or more inputs — for example:

• Temperature in a room depends on position: $T(x,y,z)$T(x,y,z)
• Profit depends on cost and selling price: $P(c,s)$P(c,s)

🔹 2️⃣ Partial Derivatives

Partial derivatives measure how a function changes with respect to one variable while keeping the others constant.

Example:$$f(x,y) = x^2y + 3y$$f(x,y)=x2y+3y $$\frac{\partial f}{\partial x} = 2xy,\quad \frac{\partial f}{\partial y} = x^2 + 3$$∂x∂f​=2xy,∂y∂f​=x2+3

Useful in:

• Multivariable optimization
• Physics (heat flow, waves)
• Economics (marginal cost, marginal profit)

🔹 3️⃣ Exact and Inexact Differentials

A differential expression of the form:$$M(x,y)dx + N(x,y)dy = 0$$M(x,y)dx+N(x,y)dy=0

is called exact if there exists a function $\phi(x,y)$ϕ(x,y) such that:$$\frac{\partial \phi}{\partial x}=M \quad \text{and} \quad \frac{\partial \phi}{\partial y}=N$$∂x∂ϕ​=Mand∂y∂ϕ​=N

Condition for exactness:$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$∂y∂M​=∂x∂N​

If this condition does not hold → the differential is called inexact.

🔹 4️⃣ Integrating Factor (IF)

Sometimes an inexact differential can be made exact by multiplying by a suitable function $\mu(x,y)$μ(x,y), called Integrating Factor:$$\mu(x,y)\big[M(x,y)dx + N(x,y)dy\big] = 0$$μ(x,y)[M(x,y)dx+N(x,y)dy]=0

Simple Illustration

$$(2xy + y^2)\,dx + (x^2 + 2xy)\,dy = 0$$(2xy+y2)dx+(x2+2xy)dy=0

Here, after checking partial derivatives, we try a suitable integrating factor depending on $x$x or $y$y.
Once exact, we integrate to find the potential function $\phi(x,y)$ϕ(x,y).

🔹 5️⃣ Constrained Maximization using Lagrange Multipliers

Used to find maxima and minima of functions subject to a constraint.

Let:$$f(x,y) \quad \text{(objective function)}, \quad g(x,y)=0 \quad \text{(constraint)}$$f(x,y)(objective function),g(x,y)=0(constraint)

We define:$$\mathcal{L}(x,y,\lambda) = f(x,y) – \lambda g(x,y)$$L(x,y,λ)=f(x,y)−λg(x,y)

Solve:$$\frac{\partial \mathcal{L}}{\partial x}=0,\; \frac{\partial \mathcal{L}}{\partial y}=0,\; \frac{\partial \mathcal{L}}{\partial \lambda}=0$$∂x∂L​=0,∂y∂L​=0,∂λ∂L​=0

Applications:

• Maximum profit subject to budget limitations
• Designing objects to minimize cost
• Optimization in Physics & Machine Learning

✨ Vector Algebra — Recap

Vectors denote quantities having magnitude and direction like force, velocity, position, etc.

🔹 Properties of Vectors under Rotations

When a vector is rotated, its length (magnitude) does not change.
Vector equations remain valid under rotation — an important property in physics.

🔹 Scalar Product (Dot Product)

$$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta$$A⋅B=∣A∣∣B∣cosθ

Represents:

• Work done by a force
• Projection of one vector on another

It remains invariant under rotations.

🔹 Vector Product (Cross Product)

$$\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin\theta \,\hat{n}$$A×B=∣A∣∣B∣sinθn^

It gives a vector perpendicular to both $\vec{A}$A and $\vec{B}$B.
Used in torque, rotational motion, magnetic forces, etc.

🔹 Scalar Triple Product

$$\vec{A} \cdot (\vec{B} \times \vec{C})$$A⋅(B×C)

Interpretation:

• Represents volume of the parallelepiped formed by the vectors

If the result = 0 ⇒ vectors are coplanar.

🔹 Vector Triple Product

$$\vec{A} \times (\vec{B} \times \vec{C}) = \vec{B}(\vec{A}\cdot\vec{C}) – \vec{C}(\vec{A}\cdot\vec{B})$$A×(B×C)=B(A⋅C)−C(A⋅B)

Useful in mechanics and electromagnetism.

🔹 Scalar and Vector Fields

They help describe flow, forces, and energy distributions in space.

📌 Conclusion

Calculus-II builds the foundation for advanced subjects like:

• Fluid dynamics
• Thermodynamics
• Optimization and Machine Learning
• Electromagnetism
• Economics and Data Science

By mastering partial derivatives, constrained optimization, and vector algebra, students gain strong analytical skills for real-world mathematical modeling.