Vector Calculus · Engineering Mathematics · GATE ECE

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Marks 1

GATE ECE 2024
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 \...
GATE ECE 2017 Set 2
The smaller angle (in degrees) between the planes $$x+y+z=1$$ and $$2x-y+2z=0$$ is ________.
GATE ECE 2015 Set 1
A vector $$\overrightarrow P $$ is given by $$\,\,\overrightarrow P = {x^3}y\overrightarrow a {}_x - {x^2}{y^2}\overrightarrow a {}_y - {x^2}yz\overr...
GATE ECE 2014 Set 4
The magnitude of the gradient for the function $$f\left( {x,y,z} \right) = {x^2} + 3{y^2} + {z^3}\,\,$$ at the point $$(1,1,1)$$ is _________.
GATE ECE 2014 Set 4
The directional derivative of $$f\left( {x,y} \right) = {{xy} \over {\sqrt 2 }}\left( {x + y} \right)$$ at $$(1, 1)$$ in the direction of the unit v...
GATE ECE 2014 Set 2
If $$\,\overrightarrow r = x\widehat a{}_x + y\widehat a{}_y + z\widehat a{}_z\,\,\,\,$$ and $$\,\left| {\overrightarrow r } \right| = r,$$ then di...
GATE ECE 2013
Consider a vector field $$\overrightarrow A \left( {\overrightarrow r } \right).$$ The closed loop line integral $$\oint {\overrightarrow A \bullet \...
GATE ECE 2011
Consider a closed surface $$'S'$$ surrounding a volume $$V.$$ If $$\overrightarrow r $$ is the position vector of a point inside $$S$$ with $$\overrig...
GATE ECE 2005
$$\nabla \times \left( {\nabla \times P} \right)\,\,$$ where $$P$$ is a vector is equal to
GATE ECE 1993
If the linear velocity $${\overrightarrow V }$$ is given by $$\overrightarrow V = {x^2}y\overrightarrow i + xyz\overrightarrow j - y{z^2}\overright...

Marks 2

GATE ECE 2024
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_...
GATE ECE 2017 Set 1
If the vector function $$\,\,\overrightarrow F = \widehat a{}_x\left( {3y - k{}_1z} \right) + \widehat a{}_y\left( {k{}_2x - 2z} \right) - \widehat ...
GATE ECE 2017 Set 1
Let $$\,\,\,{\rm I} = \int_c {\left( {2z\,dx + 2y\,dy + 2x\,dz} \right)} \,\,\,\,$$ where $$x, y, z$$ are real, and let $$C$$ be the straight line se...
GATE ECE 2016 Set 2
Suppose $$C$$ is the closed curve defined as the circle $$\,\,{x^2} + {y^2} = 1\,\,$$ with $$C$$ oriented anti-clockwise. The value of $$\,\,\oint {\...
GATE ECE 2014 Set 4
Given $$\,\,\overrightarrow F = z\widehat a{}_x + x\widehat a{}_y + y\widehat a{}_z.\,\,$$ If $$S$$ represents the portion of the sphere $${x^2} + ...
GATE ECE 2013
The divergence of the vector field $$\,\overrightarrow A = x\widehat a{}_x + y\widehat a{}_y + z\widehat a{}_z\,\,$$ is
GATE ECE 2012
The direction of vector $$A$$ is radially outward from the origin, with $$\left| A \right| = K\,{r^n}$$ where $${r^2} = {x^2} + {y^2} + {z^2}$$ and...
GATE ECE 2010
If $$\overrightarrow A = xy\,\widehat a{}_x + {x^2}\widehat a{}_y\,\,$$ then $$\,\,\oint {\overrightarrow A .d\overrightarrow r \,\,} $$ over the pat...
GATE ECE 2009
If a vector field$$\overrightarrow V $$ is related to another field $$\overrightarrow A $$ through $$\,\overrightarrow V = \nabla \times \overright...
GATE ECE 2008
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} \righ...
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