Q: Which of the following is NOT a necessary condition for a vector field $\vec{F}$ to be conservative?

The field is divergence-free ($\nabla \cdot \vec{F} = 0$).

Question 1

Which of the following best describes the gradient of a scalar field $f(x, y, z)$ in three dimensions?

Accepted Answer

It is a vector pointing in the direction of the greatest increase of $f$.

Question 2

Given a scalar field $f(x, y) = x^2 + y^2$, what is the value of the gradient $
abla f$ at the point $(1, 2)$?

Accepted Answer

$
abla f = (2, 4)$

Question 3

Which of the following is NOT a necessary condition for a vector field $\vec{F}$ to be conservative?

Accepted Answer

The field is divergence-free ($
abla \cdot \vec{F} = 0$).

Question 4

A particle moves in a potential field $V(x, y, z)$. Which equation gives the force experienced by the particle?

Accepted Answer

$\vec{F} = -
abla V$

Question 5

Which of the following statements about Lagrange multipliers is correct?

Accepted Answer

They are used to find extrema of a function subject to constraints.

Question 6

Evaluate the line integral $\int_C \vec{F} \cdot d\vec{r}$ for $\vec{F} = (y, x)$ along the path from $(0,0)$ to $(1,1)$, where $C$ is the straight line segment.

Accepted Answer

1

Question 7

Which theorem relates the surface integral of the curl of a vector field over a surface $S$ to the line integral of the field around the boundary curve $C$?

Accepted Answer

Stokes' Theorem

Question 8

Given the vector field $\vec{F} = (x, y, z)$, what is the divergence $
abla \cdot \vec{F}$?

Accepted Answer

$3$

Question 9

Which coordinate system is most appropriate for solving problems with spherical symmetry?

Accepted Answer

Spherical coordinates

Question 10

A heat equation in one dimension is given by $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$. What type of equation is this?

Accepted Answer

Partial Differential Equation (PDE)

Question 11

Which of the following is a correct application of the method of separation of variables?

Accepted Answer

Assume $u(x, t) = X(x)T(t)$ for the heat equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$.

Question 12

Which of the following is a boundary condition for a PDE describing heat flow in a rod of length $L$?

Accepted Answer

$u(0, t) = 0$

Question 13

Which of the following is a correct statement about the Laplacian operator $
abla^2$?

Accepted Answer

It is the sum of the second partial derivatives with respect to each spatial variable.

Question 14

Which of the following is an example of an initial value problem for an ordinary differential equation?

Accepted Answer

$\frac{dy}{dt} = ky$, $y(0) = y_0$

Question 15

Which of the following is a correct statement about the use of Green's Theorem?

Accepted Answer

It relates a line integral around a simple closed curve to a double integral over the region it encloses.

Question 16

Which of the following is a plausible application of ordinary differential equations in real life?

Accepted Answer

Modeling the spread of an epidemic using the SIR model.

Question 17

For the vector field $\vec{F} = (y, -x)$, what is the curl $
abla 	imes \vec{F}$ in two dimensions?

Question 18

Which of the following is a correct statement about the total derivative of a function $f(x, y)$?

Accepted Answer

$df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$

Question 19

Which of the following is a correct statement about the use of Lagrange multipliers in optimization problems?

Accepted Answer

They allow us to find extrema of a function subject to equality constraints.

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