GATE ECE 2024 | Vector Calculus Question 2 | Engineering Mathematics

GATE ECE 2024

MCQ (More than One Correct Answer)

-1.33

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?

For every closed path Γ, we have $\oint\limits_Γ (F_1 dx + F_2 dy + F_3 dz) = 0$.

There exists a differentiable scalar function $f(x, y, z)$ such that $F_1 = \frac{\partial f}{\partial x}$, $F_2 = \frac{\partial f}{\partial y}$, $F_3 = \frac{\partial f}{\partial z}$.

$\frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} = 0$.

$\frac{\partial F_3}{\partial y} = \frac{\partial F_2}{\partial z}$, $\frac{\partial F_1}{\partial z} = \frac{\partial F_3}{\partial x}$, $\frac{\partial F_2}{\partial x} = \frac{\partial F_1}{\partial y}$.

GATE ECE 2017 Set 1

MCQ (Single Correct Answer)

-0.6

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

$$0.3, -2.5, 0.5$$

$$0.0, 3.0, 2.0$$

$$0.3, 0.33, 0.5$$

$$4.0, 3.0, 2.0$$

GATE ECE 2017 Set 1

Numerical

-0

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 ___________.

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GATE ECE 2016 Set 2

Numerical

-0

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 _______.

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