Midterm 4 Practice

Conceptual Questions!

If F is a vector field, then div F is a vector field. True or False

If F is a vector field, then div F is a vector field. True or False

False: div F results in a scalar function

If F is a vector field, then curl(F) is a vector field.

True: curl(F) results in a vector function

If f has continuous partial derivatives of all orders on ℝ3, then div(curl(∇f)) = 0.

True:

If f has continuous partial derivatives of all orders on ℝ3 and C is any circle, then

∫∇f · dr = 0.

TRUE: if function is continuous and derivatives exist then line integral in a closed path is zero

If F = P i + Q j and Py = Qx in an open region D, then F is conservative.

FALSE: To satisfy the conservative property, the region D should be open AND simply connected

FALSE

If F and G are vector fields and div F = div G, then F = G

FALSE: Check this counter-example

The work done by a conservative force field in moving a particle around a closed path is zero.

TRUE: A force is conservative exactly when the work it does on an object is zero for every possible closed path the object can take.

If F and G are vector fields, then curl(F + G) = curlF + curlG

TRUE: Try an example!

If F and G are vector fields, then curl(F · G) = curlF · curlG

FALSE: Intuitively, you can see this will be false because F · G = scalar. curl is defined for vectors only

If S is a sphere and F is a constant vector field, then ∫∫s F · ndS = 0

TRUE: ∫∫s F · ndS = ∫∫∫v div F dudydz if F is constant then div F = 0 therefore, ∫∫d F · ndS = 0

There is a vector field F such that curl(F) = xi + yj + zk

FALSE: Every vector function G satisfies the property: div(curlF) = 0 if curl(F) = xi + yj + zk, then div(curl(F)) = 3 Since this results in a non zero scalar, the statement is false

The area of the region bounded by the positively oriented, piecewise smooth, simple closed curve C is A= - ∮c ydx

TRUE: The area pf the region bounded by positively oriented, piecewise smooth, simply closed curve C can be represented with A= ∮c ydx and A= ∮c ydx = - ∮c ydx

is curl(f) a ...

a) scalar field b) vector field c) not meaningful

c) not meaningful

is grad(f) a ...

a) scalar field b) vector field c) not meaningful

b) vector field

(F is a vector function) is div(F) a ...

a) scalar field b) vector field c) not meaningful

a) scalar field

is curl(grad(f)) a ...

a) scalar field b) vector field c) not meaningful

b) vector field

is grad(F) a ...

a) scalar field b) vector field c) not meaningful

c) not meaningful

is grad(div(F) a ...

a) scalar field b) vector field c) not meaningful

c) vector field

is div(grad(f)) a ...

a) scalar field b) vector field c) not meaningful

a) scalar field

is grad(div(f)) a ...

a) scalar field b) vector field c) not meaningful

c) not meaningful

is curl(curl(F)) a ...

a) scalar field b) vector field c) not meaningful

b) vector field

is div(div(F)) a ...

a) scalar field b) vector field c) not meaningful

c) not meaningful

is (grad(f)) x (div(F)) a ...

a) scalar field b) vector field c) not meaningful

c) not meaningful

is div(curl(grad(f))) a ...

a) scalar field b) vector field c) not meaningful

a) scalar field

If all the component functions of F have continuous partials, then F will be conservative if

curl(F) = 0

A vector field G exists in R^3 if...

div(curl G) = 0

Fundamental Theorem of Line Integral