Midterm 4 Practice

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Conceptual Questions!

Pre-Calculus

Vectors

University/Undergrad

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28 Terms

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

False: div F results in a scalar function

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

True: curl(F) results in a vector function

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

True:

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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

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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

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FALSE

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

FALSE: Check this counter-example

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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.

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

TRUE:
Try an example!

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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

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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

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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

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

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

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