Recent questions tagged calculus

Given that $\vec{F}(x, y, z)=\sin (y) \hat{x}+\cos (x) \hat{y}+5 \hat{z}$, the integral $\oint \oint _{S} \vec{F}(x, y, z) \cdot \overrightarrow{d s}$ over the unit spher...

Let $C$ be a clockwise oriented closed curve in the complex plane defined by $\mid z \mid =1$. Further, let $f(z)=j z$ be a complex function, where $j=\sqrt{-1}$. Then, $...

​​​​Consider a vector $\bar{u}=2 \hat{x}+\hat{y}+2 \hat{z}$, where $\hat{x}, \hat{y}, \hat{z}$ represent unit vectors along the coordinate axes $x, y, z$ respectively. Th...

​​​​Let $f(t)$ be a real-valued function whose second derivative is positive for $-\infty < t < \infty.$ Which of the following statements is/are always true?$f(t)$ has a...

​​​​​Consider the function $f(t)=(\max (0, t))^2$ for $-\infty<t<\infty$, where $\max (a, b)$ denotes the maximum of $a$ and $b$. Which of the following statements is/are...

The closed curve shown in the figure is described by $r=1+\cos \theta,$ where $r=\sqrt{x^{2}+y^{2}} ; x=r \cos \theta, y=r \sin \theta$ The magnitude of the line integra...

A quadratic function of two variables is given as $$ f\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{2}^{2}+3 x_{1}+3 x_{2}+x_{1} x_{2}+1$$The magnitude of the maximum rate of...

Let, $f\left ( x,y, z \right ) = 4x^{2} + 7xy + 3xz^{2}$. The direction in which the function $\text{f(x, y, z)}$ increases most rapidly at point $P = (1,0,2)$ is$20\hat{...

Let $R$ be a region in the first quadrant of the $xy$ plane enclosed by a closed curve $C$ considered in counter-clockwise direction. Which of the following expressions d...

Let $\vec{E}\left ( x, y, z \right ) = 2x^{2}\hat{i} + 5y\hat{j} + 3z\hat{k}$. The value of $\iiint_{V} \left ( \vec{\triangledown} . \vec{E} \right )dV$, where $V$ is th...

Suppose the circles $x^{2}+y^{2}=1$ and $\left ( x-1\right )^{2}+\left ( y-1 \right )^{2}=r^{2}$ intersect each other orthogonally at the point $\left ( u,v \right )$. Th...

In the open interval $\left ( 0,1 \right )$, the polynomial $p\left ( x \right) =x^{4}-4x^{3}+2$ hastwo real rootsone real rootthree real rootsno real roots

Let $\left ( -1 -j \right ), \left ( 3 -j \right ), \left ( 3 + j \right )$ and $\left ( -1+ j \right )$ be the vertices of a rectangle $C$ in the complex plane. Assuming...

Which of the following is true for all possible non-zero choices of integers $m,n;m\neq n,$ or all possible non-zero choices of real numbers $p,q;p\neq q,$ as applicable?...

The vector function expressed by$$F=a_{x}\left ( 5y-k_{1} z\right )+a_{y}\left ( 3z+k_{2}x \right )+a_{z}\left ( k_{3} y-4x\right )$$represents a conservative field, wher...

Let $a_{x}$ and $a_{y}$ be unit vectors along $x$ and $y$ directions, respectively. A vector function is given by $$F=a_{x}y-a_{y}x$$The line integral of the above functi...

If $f=2x^{3}+3y^{2}+4z$, the value of line integral $\int_{c} \text{grad}f \cdot d \textbf{r}$ evaluated over contour $C$ formed by the segments $(-3,-3,2)\rightarrow(2,-...

A periodic function $f(t)$, with a period of $2 \pi$, is represented as its Fourier series, $$f(t) = a_0 + \sum_{n=1}^{\infty }a_n \cos nt + \sum_{n=1}^{\infty} b_n \sin ...

If $\textbf{A}= 2x \textbf{i} + 3y \textbf{j} +4z \textbf{k}$ and $u=x^2+y^2+z^2$, then $\text{div} \big(u \textbf{A} \big)$ at $(1,1,1)$ is _______

The direction of vector $\textbf{A}$ is radically outward from the origin, with $\mid \textbf{A} \mid k r ^n$ where $r^2=x^2+y^2+z^2$ and $k$ is a constant. The value of ...