Review MAT 235
Mohammad El Smaily
Department of Mathematics,
University of Toronto
March 27, 2014
M. El Smaily (Math. Toronto)
MAT 235 Review
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Geometry in space
Problem
Find the equation of line which is normal to −x + 2y + z = 5 and passes
through (2, 1, −1)
a) −(x − 2) + 2(y − 1) + (z + 1) = 0
b) x = 2t − 1, y = t + 2, z = −t + 1
c) hx, y , zi = h2, 1, −1i + th2, 1, −1i
d)
e)
x−2
2
x−1
2
=
=
y −1
−4
y −2
2
=
=
z+1
−2
z−1
−1
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Critical points, Min-Max
Problem
Let f (x, y ) = x 2 − y 2 on R2 . Choose a wrong statement.
a) (0, 0) is the only critical point of f .
b) There is no local maximum of f .
c) fxx = 2.
d) f has a local minimum at (0, 0).
e) fxy = 0.
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Problem
Let f (x, y ) = x 2 + y 2 on {(x, y ) ∈ R2 , |x| + |y | = 1}. Choose one correct
statement below.
a) The absolute maximum of f is
1
2
and the absolute minimum of f is 0.
b) The absolute maximum of f is 1 and there is no absolute minimum.
c) The absolute minimum of f is 0 and there is no absolute maximum.
d) There is no extreme values.
e) The absolute maximum of f is 1 and the absolute minimum of f is 12 .
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Line Integrals (0)
Problem
R
Compute C f ∇f · dr where f (x, y , z) = xzcos(x 2 + y 2 ) and C is the
intersection of the cylinder x 2 + y 2 = 1 and x + y + 2z = 2.
a) 0.
b) π.
c) −2π.
d) 2π.
e) None of the above
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Line Integrals (1)
Problem
R
Evaluate C ∇f · dr where f (x, y , z) = xy + y 2 + z and C is the
parametric curve hcos t, sin t, ti, 0 ≤ t ≤ π (the curve starts from t = 0)
a) −1 + π
b) 2π
c) π
d) 1 + π
e) 1.
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Line Integrals (2)
Problem
R
Without calculation or application of any theorems, determine if C F · dr
is positive, negative, or zero. F is the vector field pictured below. C is the
circle traced out as indicated
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Solution
Recall that dr corresponds to an very small tangent vector along the path
C.
It appears that F is of constant magnitude, pointing in the i + j direction.
If two vectors point along the same general direction, their dot product is
positive.
Along the bottom-right semicircle, F · dr is positive because dr points
along F. Similarly, F · dr is negative along the top-left semicircle because
dr points against F .
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Solution-Continued
It appears that
R F · dr takes on positive and negative values equally, so we
suspect that C F · dr = 0
In order to verify that the line integral is zero, we break it into many pieces
and see if we can see some cancellation with a piece on the opposite side
of the circle. We draw two opposing pieces.
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MAT 235 Review
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Observing that the vector dr on the bottom is of the same magnitude but
opposite direction as the dr vector on the top, we can see that the
contribution of F · dr from the bottom exactly cancels the contribution
from the top. Because the line integral is the sum of all segments forming
the circle,Rand because each segment cancels the value of its opposite
segment, C F · dr = 0.
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MAT 235 Review
March 27, 2014
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Line Integrals (3)
R
Without calculation, determine if C F · dr is positive, negative, or zero. F
is the vector field pictured below. C is the red circle traced out clockwise.
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Line Integrals (4)
Problem
Let F (x, y ) = h2, 3i.
R Suppose C is a curve connecting (0, 0) to (1, 1).
Does the value of C F · dr depend on the shape of the curve C ? If not,
find the value of the integral.
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Visualization
Problem
Sketch the region in 2d satisfied by (1, 1) · x ≥ 2.
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Parametric Surfaces
Problem
Let S be given by the vector valued function r (u, v ) = hu cos v , u sin v , v i.
Find the tangent plane to the surface at r (1, 4).
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