Partial Differentiation - GATE Study Material

Introduction
Partial differentiation is a crucial topic in the Engineering Mathematics section of the Graduate Aptitude Test in Engineering (GATE). It forms the foundation for understanding multivariable calculus, which is essential for various engineering disciplines including Mechanical, Civil, Electrical, and Chemical Engineering. In GATE, questions from this area typically test concepts like finding first and second-order derivatives, the chain rule, Euler's theorem, and applications such as maxima and minima.

1. Basic Concepts and Definitions

Unlike ordinary differentiation, where a function depends on a single variable ($y = f(x)$), partial differentiation deals with functions of multiple variables ($z = f(x, y)$). When we find the partial derivative of a function with respect to one variable, we treat all other variables as constants.

• First Order Partial Derivatives: For a function $z = f(x, y)$, the partial derivative with respect to $x$ is denoted by $\frac{\partial z}{\partial x}$ or $z_x$. It measures the rate of change of $z$ as $x$ changes, keeping $y$ constant.
• Geometrical Interpretation: $\frac{\partial z}{\partial x}$ represents the slope of the tangent to the curve formed by the intersection of the surface $z = f(x, y)$ and a plane $y = \text{constant}$.

2. Important Rules and Theorems

Chain Rule for Partial Differentiation

If $u$ is a function of $x$ and $y$, i.e., $u = u(x, y)$, and $x$ and $y$ are themselves functions of $t$ (i.e., $x = x(t)$ and $y = y(t)$), then the total derivative of $u$ with respect to $t$ is given by:

Homogeneous Functions and Euler's Theorem

This is a high-weightage topic in GATE. A function $f(x, y)$ is said to be homogeneous of degree $n$ if for any scalar $t$, $f(tx, ty) = t^n f(x, y)$.

Euler's Theorem: If $f(x, y)$ is a homogeneous function of degree $n$, then:

For higher derivatives, the theorem can be extended. This theorem is often used to simplify differential equations or verify solutions quickly.

3. Higher Order Partial Derivatives

Just as we have second derivatives in single-variable calculus, we have second-order partial derivatives. For $z = f(x, y)$, there are four possibilities:

• $\frac{\partial}{\partial x}(\frac{\partial z}{\partial x}) = \frac{\partial^2 z}{\partial x^2}$ (Differentiate w.r.t $x$ twice)
• $\frac{\partial}{\partial y}(\frac{\partial z}{\partial y}) = \frac{\partial^2 z}{\partial y^2}$ (Differentiate w.r.t $y$ twice)
• $\frac{\partial}{\partial y}(\frac{\partial z}{\partial x}) = \frac{\partial^2 z}{\partial y \partial x}$
• $\frac{\partial}{\partial x}(\frac{\partial z}{\partial y}) = \frac{\partial^2 z}{\partial x \partial y}$

Schwarz's Theorem (Clairaut's Theorem): If the function $f(x, y)$ and its partial derivatives are continuous, then the order of differentiation does not matter for mixed derivatives:

4. Applications: Maxima and Minima

Finding the extreme values of a function of two variables is a standard problem in GATE. To find the maxima or minima of $z = f(x, y)$:

1. Find the first partial derivatives $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.
2. Solve $\frac{\partial z}{\partial x} = 0$ and $\frac{\partial z}{\partial y} = 0$ simultaneously to get stationary points $(a, b)$.
3. Calculate second-order derivatives at the stationary point:
   $A = \frac{\partial^2 z}{\partial x^2}$ at $(a, b)$
   $B = \frac{\partial^2 z}{\partial x \partial y}$ at $(a, b)$
   $C = \frac{\partial^2 z}{\partial y^2}$ at $(a, b)$

5. Method of Lagrange Multipliers (Constrained Optimization)

Sometimes, we need to find the extrema of $f(x, y)$ subject to a constraint $\phi(x, y) = 0$. The method of Lagrange multipliers converts this constrained problem into an unconstrained one using a new variable $\lambda$ (lambda).

We define a new function:

The necessary conditions for extremum are:

Solving these three equations yields the values of $x, y$ at which the extremum occurs under the given constraint. This is frequently asked in GATE to test conceptual understanding beyond simple unconstrained optimization.

6. Total Derivatives

If $z = f(x, y)$ where $x = \phi(t)$ and $y = \psi(t)$, then $z$ is ultimately a function of $t$. The total derivative $\frac{dz}{dt}$ connects partial derivatives with ordinary derivatives:

Dividing by $dt$ gives the total derivative formula mentioned in the Chain Rule section. A crucial distinction to remember is that $\frac{\partial z}{\partial x}$ implies $y$ is constant, whereas $\frac{dz}{dx}$ implies $y$ might vary with $x$ (if there is a relation between $x$ and $y$).

7. Jacobian Determinant

While slightly more advanced, the Jacobian appears in coordinate transformations and change of variables in multiple integrals. If $u$ and $v$ are functions of $x$ and $y$:

Properties like $J \cdot J' = 1$ (where $J'$ is the Jacobian of inverse transformation) are useful identities for objective problems.

Conclusion

To excel in Partial Differentiation for GATE, focus on speed and accuracy in calculating derivatives. Master Euler's theorem for homogeneous functions, as it provides quick solutions to complex-looking questions. Also, practice problems on constrained optimization using Lagrange multipliers, as this concept distinguishes top scorers. Ensure you are comfortable with the notation and understand the geometric meaning of the derivatives to tackle application-based problems effectively.