Calculus Quick Notes for GATE Electrical Engineering (EE)

Mean Value Theorems

Mean Value Theorems

Rolle’s Theorem: If \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and \(f(a) = f(b)\), then \(\exists c \in (a,b)\) such that \(f'(c) = 0\).

Lagrange’s Mean Value Theorem: If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then \(\exists c \in (a,b)\) such that:

\[f'(c) = \frac{f(b) - f(a)}{b - a}\]

Cauchy’s Mean Value Theorem: If \(f\) and \(g\) are continuous on \([a,b]\), differentiable on \((a,b)\), and \(g'(x) \neq 0\) on \((a,b)\), then \(\exists c \in (a,b)\) such that:

\[\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}\]

Theorems of Integral Calculus

Fundamental Theorems of Calculus

First Fundamental Theorem: If \(F(x) = \int_a^x f(t) dt\) and \(f\) is continuous, then \(F'(x) = f(x)\).

Second Fundamental Theorem:

\[\int_a^b f(x) dx = F(b) - F(a)\]
where \(F'(x) = f(x)\).

Integration by Parts:

\[\int u dv = uv - \int v du\]

Substitution Rule:

\[\int f(g(x))g'(x) dx = \int f(u) du\]
where \(u = g(x)\)

Evaluation of Integrals

Definite and Improper Integrals

Properties of Definite Integrals:

  • \(\int_a^b f(x) dx = -\int_b^a f(x) dx\)

  • \(\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx\)

  • \(\int_0^a f(x) dx = \int_0^a f(a-x) dx\)

Improper Integrals:

  • Type I: \(\int_a^{\infty} f(x) dx = \lim_{t \to \infty} \int_a^t f(x) dx\)

  • Type II: \(\int_a^b f(x) dx\) where \(f\) has discontinuity at \(c \in [a,b]\)

    \[= \lim_{\epsilon \to 0^-} \int_a^{c+\epsilon} f(x) dx + \lim_{\epsilon \to 0^+} \int_{c+\epsilon}^b f(x) dx\]

Common Integral Forms

Standard Forms:

\[\begin{aligned} \int \frac{dx}{x^2 + a^2} &= \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \\ \int \frac{dx}{x^2 - a^2} &= \frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C \\ \int \frac{dx}{\sqrt{a^2 - x^2}} &= \sin^{-1}\left(\frac{x}{a}\right) + C \\ \int \frac{dx}{\sqrt{x^2 + a^2}} &= \ln|x + \sqrt{x^2 + a^2}| + C \end{aligned}\]

Gamma Function: \(\Gamma(n) = \int_0^{\infty} t^{n-1} e^{-t} dt\)

  • \(\Gamma(n+1) = n\Gamma(n)\)

  • \(\Gamma(n) = (n-1)!\) for positive integers

Partial Derivatives

Partial Derivatives

Definition: For \(z = f(x,y)\):

\[\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}\]

Higher Order Partial Derivatives:

\[f_{xx} = \frac{\partial^2 f}{\partial x^2}, \quad f_{xy} = \frac{\partial^2 f}{\partial x \partial y}, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2}\]

Schwarz’s Theorem: If \(f_{xy}\) and \(f_{yx}\) are continuous, then \(f_{xy} = f_{yx}\).

Chain Rule: If \(z = f(x,y)\), \(x = g(t)\), \(y = h(t)\):

\[\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}\]

Maxima and Minima

Maxima and Minima

Critical Points: Points where \(\nabla f = 0\) or undefined.

Second Derivative Test: For \(f(x,y)\) at critical point \((a,b)\): Let \(D = f_{xx}f_{yy} - (f_{xy})^2\) at \((a,b)\)

  • If \(D > 0\) and \(f_{xx} > 0\): local minimum

  • If \(D > 0\) and \(f_{xx} < 0\): local maximum

  • If \(D < 0\): saddle point

  • If \(D = 0\): test inconclusive

Lagrange Multipliers: To optimize \(f(x,y)\) subject to \(g(x,y) = 0\):

\[\nabla f = \lambda \nabla g\]
Solve: \(f_x = \lambda g_x\), \(f_y = \lambda g_y\), \(g(x,y) = 0\)

Multiple Integrals

Double and Triple Integrals

Double Integral:

\[\iint_R f(x,y) \, dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \, dx\]

Change of Variables: For \(x = x(u,v)\), \(y = y(u,v)\):

\[\iint_R f(x,y) \, dx \, dy = \iint_S f(x(u,v), y(u,v)) \left|\frac{\partial(x,y)}{\partial(u,v)}\right| du \, dv\]

Polar Coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\)

\[\iint_R f(x,y) \, dx \, dy = \iint_S f(r\cos\theta, r\sin\theta) \, r \, dr \, d\theta\]

Triple Integral:

\[\iiint_E f(x,y,z) \, dV = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z) \, dz \, dy \, dx\]

Coordinate Systems for Triple Integrals

Cylindrical Coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\), \(z = z\)

\[dV = r \, dr \, d\theta \, dz\]

Spherical Coordinates: \(x = \rho\sin\phi\cos\theta\), \(y = \rho\sin\phi\sin\theta\), \(z = \rho\cos\phi\)

\[dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta\]

Applications:

  • Volume: \(V = \iiint_E dV\)

  • Mass: \(m = \iiint_E \rho(x,y,z) dV\)

  • Center of mass: \(\bar{x} = \frac{1}{m}\iiint_E x\rho(x,y,z) dV\)

Fourier Series

Fourier Series

Fourier Series Representation:

\[f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\frac{n\pi x}{L} + b_n \sin\frac{n\pi x}{L}\right)\]

Fourier Coefficients: For period \(2L\):

\[\begin{aligned} a_0 &= \frac{1}{L} \int_{-L}^{L} f(x) dx \\ a_n &= \frac{1}{L} \int_{-L}^{L} f(x) \cos\frac{n\pi x}{L} dx \\ b_n &= \frac{1}{L} \int_{-L}^{L} f(x) \sin\frac{n\pi x}{L} dx \end{aligned}\]

Special Cases:

  • Even function: \(b_n = 0\) (cosine series)

  • Odd function: \(a_n = 0\) (sine series)

Convergence and Properties

Dirichlet Conditions: Fourier series converges if:

  • \(f\) is bounded

  • \(f\) has finite number of maxima and minima

  • \(f\) has finite number of discontinuities

Convergence: At discontinuity points:

\[f(x_0^+) + f(x_0^-) = 2S\]
where \(S\) is the sum of the series.

Parseval’s Identity:

\[\frac{1}{L}\int_{-L}^{L} [f(x)]^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty}(a_n^2 + b_n^2)\]

Vector Calculus

Vector Identities

Basic Operations:

\[\begin{aligned} \nabla f &= \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) \\ \nabla \cdot \mathbf{F} &= \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \\ \nabla \times \mathbf{F} &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} \end{aligned}\]

Important Identities:

  • \(\nabla \times (\nabla f) = \mathbf{0}\)

  • \(\nabla \cdot (\nabla \times \mathbf{F}) = 0\)

  • \(\nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - \nabla^2\mathbf{F}\)

  • \(\nabla(fg) = f\nabla g + g\nabla f\)

Directional Derivatives

Definition: Directional derivative of \(f\) at point \(P\) in direction \(\mathbf{u}\):

\[D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}\]
where \(\mathbf{u}\) is a unit vector.

Maximum Rate of Change:

  • Direction: \(\mathbf{u} = \frac{\nabla f}{|\nabla f|}\)

  • Maximum rate: \(|\nabla f|\)

Gradient Properties:

  • \(\nabla f\) is perpendicular to level curves/surfaces

  • \(\nabla f\) points in direction of maximum increase

  • \(|\nabla f|\) gives the maximum rate of change

Line Integrals

Line Integrals

Line Integral of Scalar Field:

\[\int_C f(x,y) ds = \int_a^b f(x(t), y(t)) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt\]

Line Integral of Vector Field:

\[\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(x(t), y(t)) \cdot \mathbf{r}'(t) dt\]

Conservative Vector Fields: \(\mathbf{F}\) is conservative if \(\nabla \times \mathbf{F} = \mathbf{0}\) (in 2D: \(\frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial y}\))

For conservative fields: \(\int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A)\) where \(\nabla f = \mathbf{F}\)

Green’s Theorem:

\[\oint_C (P dx + Q dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA\]

Surface and Volume Integrals

Surface Integrals

Surface Integral of Scalar Field:

\[\iint_S f(x,y,z) dS = \iint_D f(x,y,g(x,y)) \sqrt{1 + \left(\frac{\partial g}{\partial x}\right)^2 + \left(\frac{\partial g}{\partial y}\right)^2} dA\]

Surface Integral of Vector Field:

\[\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n} dS\]

Parametric Surface: \(\mathbf{r}(u,v) = (x(u,v), y(u,v), z(u,v))\)

\[d\mathbf{S} = \left|\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}\right| du dv\]

Applications:

  • Surface area: \(A = \iint_S dS\)

  • Flux: \(\Phi = \iint_S \mathbf{F} \cdot \mathbf{n} dS\)

Major Theorems

Stokes’s Theorem

Statement:

\[\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} dS\]

where \(C\) is the boundary of surface \(S\), and \(\mathbf{n}\) is the unit normal to \(S\).

Conditions:

  • \(S\) is an oriented smooth surface

  • \(C\) is a simple closed curve forming the boundary of \(S\)

  • \(\mathbf{F}\) is a vector field with continuous partial derivatives

Physical Interpretation: Circulation around a curve equals the flux of curl through the surface bounded by the curve.

Gauss’s Theorem (Divergence Theorem)

Statement:

\[\iiint_E \nabla \cdot \mathbf{F} dV = \iint_S \mathbf{F} \cdot \mathbf{n} dS\]

where \(E\) is a solid region and \(S\) is its boundary surface with outward normal \(\mathbf{n}\).

Conditions:

  • \(E\) is a simple solid region

  • \(S\) is a piecewise smooth closed surface

  • \(\mathbf{F}\) has continuous partial derivatives

Applications:

  • Fluid flow: relates flow out of region to sources/sinks inside

  • Electrostatics: Gauss’s law for electric fields

  • Heat conduction: relates heat sources to surface heat flux

Summary of Major Theorems

Fundamental Theorem for Line Integrals:

\[\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A)\]

Green’s Theorem (2D):

\[\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_D (\nabla \times \mathbf{F}) \cdot \mathbf{k} dA\]

Stokes’s Theorem (3D):

\[\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} dS\]

Divergence Theorem:

\[\iiint_E \nabla \cdot \mathbf{F} dV = \iint_S \mathbf{F} \cdot \mathbf{n} dS\]

These theorems connect different types of integrals and are fundamental to vector calculus applications.

Important Formulas Summary

  • Gradient: \(\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\)

  • Divergence: \(\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\)

  • Curl: \(\nabla \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\)

  • Laplacian: \(\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\)

  • Volume Elements:

    • Cartesian: \(dV = dx dy dz\)

    • Cylindrical: \(dV = r dr d\theta dz\)

    • Spherical: \(dV = \rho^2 \sin\phi d\rho d\phi d\theta\)