GATE ECE
Engineering Mathematics
Vector Calculus
Previous Years Questions

## Marks 1

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

## Marks 2

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