Q: Find the gradient of

By systematically calculating the vector gradient $\nabla f$ utilizing partial derivatives, and precisely evaluating it at the specified coordinate point $(3, 1, 1)$, the exact resulting vector is $7\mathbf{i} + 18\mathbf{j} - 2\mathbf{k}$.

Question 1

What is the value of integral

Accepted Answer

The Gaussian integral $\int_{0}^{\infty} e^{-x^{2}}\, dx$ is rigorously evaluated using the properties of the Gamma function by employing a substitution $t = x^2$, which elegantly yields the final exact result of $\frac{\sqrt{\pi}}{2}$.

Question 2

Write the formula of surface area of solid of revolution when the revolution …

Accepted Answer

The fundamental mathematical formula for calculating the surface area of a solid generated by revolving a continuous Cartesian curve $y = f(x)$ exactly about the horizontal x-axis is rigorously given by the definite integral $S = \int_{a}^{b} 2\pi y \sqrt{1 + (dy/dx)^2} dx$.

Question 3

What do you mean by convergence of a sequence?

Accepted Answer

In real analysis, a mathematical sequence is formally defined as convergent if its individual terms progressively and strictly approach a single, unique, and finite real number (denoted as $L$) as the index $n$ approaches infinity.

Question 4

Find whether the following series is convergent or not?

Accepted Answer

By rigorously applying the standard Limit Comparison Test using an appropriate auxiliary $p$-series, we can definitively prove that the given infinite series with the general $n$-th term $u_n = \frac{1}{(n+1)(n+2)}$ is completely convergent.

Question 5

State Parseval's theorem.

Accepted Answer

Parseval's theorem is a powerful mathematical identity in Fourier analysis that establishes a strict relationship between the average value of the square of a function, $[f(x)]^2$, and the infinite sum of the squares of its calculated Fourier coefficients.

Question 6

Find the value of $a_0$ for the function $f(x) = |x|$ in the interval $(-\pi,…

Accepted Answer

By utilizing the standard Fourier integral formula and recognizing the absolute value function $f(x) = |x|$ as a strictly even function over the symmetric interval $[-\pi, \pi]$, the Fourier coefficient $a_{0}$ is rigorously calculated to be exactly $\pi$.

Question 7

State the necessary and sufficient conditions for the minimum of a function $…

Accepted Answer

For a two-variable function $f(x,y)$, the absolute necessary conditions for an extreme point are $f_x = 0$ and $f_y = 0$, while the strict sufficient conditions confirming a local minimum are $rt - s^2 > 0$ and $r > 0$.

Question 8

Find the gradient of

Accepted Answer

By systematically calculating the vector gradient $
abla f$ utilizing partial derivatives, and precisely evaluating it at the specified coordinate point $(3, 1, 1)$, the exact resulting vector is $7\mathbf{i} + 18\mathbf{j} - 2\mathbf{k}$.

Question 9

Evaluate

Accepted Answer

The double integral $\int_{0}^{b}\int_{0}^{x} xy\, dy\, dx$ is systematically evaluated using iterated power rule integration, first with respect to $y$ and then with respect to $x$, mathematically yielding the exact final result of $\frac{b^4}{8}$.

Question 10

State the Gauss Divergence theorem.

Accepted Answer

The Gauss Divergence Theorem mathematically equates the total outward surface flux of a vector field across a closed boundary surface directly to the volume integral of the divergence of that vector field throughout the enclosed 3D region.

Question 11

Use beta and gamma functions to evaluate:

Accepted Answer

The integral evaluates to exactly $1/126$. It is rigorously solved by utilizing the fundamental symmetric properties of the Beta function, $B(m,n) = B(n,m)$, and its intrinsic mathematical relationship to the Eulerian Gamma function, $B(m,n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$.

Question 12

Expand $\sin x$ in the powers of $\left(x - \dfrac{\pi}{2}\right)$ using Tayl…

Accepted Answer

The Taylor series expansion of $f(x) = \sin x$ explicitly evaluated around the absolute point $a = \pi/2$ results in the alternating even-powered polynomial series: $1 - \frac{(x-\pi/2)^2}{2!} + \frac{(x-\pi/2)^4}{4!} - \dots$

Question 13

Find Fourier series of $x^2$ in $(-\pi,\, \pi)$, and use Parseval's identity …

Accepted Answer

By rigorously calculating the Fourier series for $f(x) = x^2$ and applying Parseval's powerful energy conservation identity, we definitively prove the absolute mathematical convergence of the infinite $p$-series $\sum_{n=1}^\infty \frac{1}{n^4}$ to exactly $\frac{\pi^4}{90}$.

Question 14

If $u = e^{xyz}$, then show that:

Accepted Answer

The third-order mixed partial derivative of the scalar function $u = e^{xyz}$ with respect to $x$, $y$, and $z$ is rigorously evaluated using successive product rule applications to yield $u_{xyz} = e^{xyz} (1 + 3xyz + x^2y^2z^2)$.

Question 15

Whether the fluid motion given by

Accepted Answer

The vector field representing the fluid motion is mathematically proven to be completely incompressible because the calculated divergence of its velocity field, $
abla \cdot \mathbf{V}$, evaluates to precisely zero across the entire mathematical space.

Question 16

Change the order of integration and hence evaluate:

Accepted Answer

By radically altering the order of integration from $dy \, dx$ to $dx \, dy$, the previously impossible integral $\int_0^1 \int_{e^x}^e \frac{1}{\log y} \, dy \, dx$ transforms into a trivial integration, elegantly yielding the exact final result of $e - 1$.

Question 17

Evaluate

Accepted Answer

The iterated triple integral $\int_1^2 \int_1^z \int_0^{yz} xyz \, dx \, dy \, dz$ is rigorously evaluated step-by-step from the innermost bound outward, ultimately yielding the exact rational fraction $9/4$, or $2.25$.

Question 18

Use beta and gamma functions to evaluate:

Accepted Answer

This complex problem requires the deep application of the Beta and Gamma functions to evaluate two distinct, improper definite integrals. Part (a) is mathematically solved using a high-order algebraic substitution to yield precisely $\frac{\pi}{3\sqrt{3}}$. Part (b) evaluates directly to $\frac{\pi}{2}$ using standard Beta formulations.

Question 19

Find the Fourier series expansion of the following periodic function with per…

Accepted Answer

This detailed problem requires the rigorous derivation of the complete Fourier series for a discontinuous square wave function. By evaluating the resulting infinite trigonometric series at a precise point of continuity, $x = \pi/2$, we mathematically deduce the famous Leibniz infinite series formula for Pi, proving exactly that $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + \dots$

Question 20

Use Lagrange's method to find the maximum and minimum distance of the point $…

Accepted Answer

This optimization problem utilizes the powerful Method of Lagrange Multipliers to strictly determine the maximum and minimum distances from a given point $(3, 4, 12)$ to the surface of a perfect unit sphere $x^2 + y^2 + z^2 = 1$. The mathematical solution rigorously proves the maximum distance is exactly 14, and the minimum is 12.

Question 21

If $u = f(r)$, where $r^{2} = x^{2} + y^{2}$, then prove that:

Accepted Answer

This derivation mathematically transforms the Cartesian Laplacian operator $
abla^2 u$ into standard 2D polar coordinates for a strictly radial function $f(r)$. By applying the multi-variable chain rule to the coordinate transformations $r = \sqrt{x^2+y^2}$, the Laplacian elegantly simplifies to $f''(r) + \frac{1}{r} f'(r)$.

Question 22

Verify Green's theorem for

Accepted Answer

This problem mathematically verifies Green's Theorem by strictly evaluating both the line integral around a closed boundary curve $C$ and the double area integral over the enclosed region $R$. Both mathematically independent calculation methods evaluate to precisely $-1/20$, perfectly verifying the theorem.

Mathematics I