Vector Calculus · Engineering Mathematics · GATE ECE

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Marks 1

1

Let $\rho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho }{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$ and $\hat{n}$ be the outward unit normal of $S$. Which of the following equations is/are equivalent to $\frac{\partial \rho }{\partial t} + \nabla \cdot(\rho u) = 0$?

GATE ECE 2024
2

Let $${v_1} = \left[ {\matrix{ 1 \cr 2 \cr 0 \cr } } \right]$$ and $${v_2} = \left[ {\matrix{ 2 \cr 1 \cr 3 \cr } } \right]$$ be two vectors. The value of the coefficient $$\alpha$$ in the expression $${v_1} = \alpha {v_2} + e$$, which minimizes the length of the error vector e, is

GATE ECE 2023
3

The rate of increase, of a scalar field $$f(x,y,z) = xyz$$, in the direction $$v = (2,1,2)$$ at a point (0,2,1) is

GATE ECE 2023
4
The smaller angle (in degrees) between the planes $$x+y+z=1$$ and $$2x-y+2z=0$$ is ________.
GATE ECE 2017 Set 2
5
A vector $$\overrightarrow P $$ is given by $$\,\,\overrightarrow P = {x^3}y\overrightarrow a {}_x - {x^2}{y^2}\overrightarrow a {}_y - {x^2}yz\overrightarrow a {}_z.\,\,\,$$ Which one of the following statements is TRUE?
GATE ECE 2015 Set 1
6
The directional derivative of $$f\left( {x,y} \right) = {{xy} \over {\sqrt 2 }}\left( {x + y} \right)$$ at $$(1, 1)$$ in the direction of the unit vector at an angle of $${\pi \over 4}$$ with $$y-$$axis, is given by ________.
GATE ECE 2014 Set 4
7
The magnitude of the gradient for the function $$f\left( {x,y,z} \right) = {x^2} + 3{y^2} + {z^3}\,\,$$ at the point $$(1,1,1)$$ is _________.
GATE ECE 2014 Set 4
8
If $$\,\overrightarrow r = x\widehat a{}_x + y\widehat a{}_y + z\widehat a{}_z\,\,\,\,$$ and $$\,\left| {\overrightarrow r } \right| = r,$$ then div $$\left( {{r^2}\nabla \left( {\ln \,r} \right)} \right) $$ = ________.
GATE ECE 2014 Set 2
9
Consider a vector field $$\overrightarrow A \left( {\overrightarrow r } \right).$$ The closed loop line integral $$\oint {\overrightarrow A \bullet \overrightarrow {dl} } $$ can be expressed as
GATE ECE 2013
10
Consider a closed surface $$'S'$$ surrounding a volume $$V.$$ If $$\overrightarrow r $$ is the position vector of a point inside $$S$$ with $$\overrightarrow n $$ the unit normal on $$'S',$$ the value of the integral GATE ECE 2011 Engineering Mathematics - Vector Calculus Question 17 English is
GATE ECE 2011
11
$$\nabla \times \left( {\nabla \times P} \right)\,\,$$ where $$P$$ is a vector is equal to
GATE ECE 2005
12
If the linear velocity $${\overrightarrow V }$$ is given by $$\overrightarrow V = {x^2}y\overrightarrow i + xyz\overrightarrow j - y{z^2}\overrightarrow k $$ then the angular velocity $$\overrightarrow W $$ at the point $$\left( {1,1, - 1} \right)$$ is _______.
GATE ECE 1993

Marks 2

1

Let $F_1$, $F_2$, and $F_3$ be functions of $(x, y, z)$. Suppose that for every given pair of points A and B in space, the line integral $\int\limits_C (F_1 dx + F_2 dy + F_3 dz)$ evaluates to the same value along any path C that starts at A and ends at B. Then which of the following is/are true?

GATE ECE 2024
2
If the vector function
$$\,\,\overrightarrow F = \widehat a{}_x\left( {3y - k{}_1z} \right) + \widehat a{}_y\left( {k{}_2x - 2z} \right) - \widehat a{}_z\left( {k{}_3y + z} \right)\,\,\,$$
is irrotational, then the values of the constants $$\,{k_1},\,{k_2}\,\,$$ and $$\,{k_3}$$ respectively, are
GATE ECE 2017 Set 1
3
Let $$\,\,\,{\rm I} = \int_c {\left( {2z\,dx + 2y\,dy + 2x\,dz} \right)} \,\,\,\,$$ where $$x, y, z$$ are real, and let $$C$$ be the straight line segment from point $$A: (0, 2, 1)$$ to point $$B: (4,1,-1).$$ The value of $${\rm I}$$ is ___________.
GATE ECE 2017 Set 1
4
Suppose $$C$$ is the closed curve defined as the circle $$\,\,{x^2} + {y^2} = 1\,\,$$ with $$C$$ oriented anti-clockwise. The value of $$\,\,\oint {\left( {x{y^2}dx + {x^2}ydx} \right)\,\,} $$ over the curve $$C$$ equals _______.
GATE ECE 2016 Set 2
5
Given $$\,\,\overrightarrow F = z\widehat a{}_x + x\widehat a{}_y + y\widehat a{}_z.\,\,$$ If $$S$$ represents the portion of the sphere $${x^2} + {y^2} + {z^2} = 1$$ for $$\,z \ge 0,$$ then $$\int\limits_s {\left( {\nabla \times \overrightarrow F .} \right)\overrightarrow {ds} \,\,} $$ is ________.
GATE ECE 2014 Set 4
6
The divergence of the vector field $$\,\overrightarrow A = x\widehat a{}_x + y\widehat a{}_y + z\widehat a{}_z\,\,$$ is
GATE ECE 2013
7
The direction of vector $$A$$ is radially outward
from the origin, with $$\left| A \right| = K\,{r^n}$$
where $${r^2} = {x^2} + {y^2} + {z^2}$$ and $$K$$ is constant.
The value of $$n$$ for which $$\nabla .A = 0\,\,$$ is
GATE ECE 2012
8
If $$\overrightarrow A = xy\,\widehat a{}_x + {x^2}\widehat a{}_y\,\,$$ then $$\,\,\oint {\overrightarrow A .d\overrightarrow r \,\,} $$ over the path shown in the figure is GATE ECE 2010 Engineering Mathematics - Vector Calculus Question 16 English
GATE ECE 2010
9
If a vector field$$\overrightarrow V $$ is related to another field $$\overrightarrow A $$ through $$\,\overrightarrow V = \nabla \times \overrightarrow A ,$$ which of the following is true?

Note: $$C$$ and $${S_C}$$ refer to any closed contour and any surface whose boundary is $$C.$$

GATE ECE 2009
10
Consider points $$P$$ and $$Q$$ in $$xy-$$plane with $$P=(1,0)$$ and $$Q=(0,1).$$ The line integral $$2\int\limits_P^Q {\left( {x\,dx + y\,dy} \right)\,\,} $$ along the semicircle with the line segment $$PQ$$ as its diameter
GATE ECE 2008
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