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Vector CalculusQuestion and Answers: Page 6

Question Number 5153 Answers: 1 Comments: 0

proof e^(iΘ) =cos(Θ)+isin(Θ)

$p r o o f e^{i Θ} = c o s (Θ) + i s i n (Θ)$

Question Number 5115 Answers: 0 Comments: 0

Given the space curve r=r(t), show that its torsion τ is given by τ=((r^. •r^(..) ×r^(...) )/(∣r^. ×r^(..) ∣^2 )). It may help to know that its curvature is numerically given by κ=((∣r^. ×r^(..) ∣)/(∣r^. ∣^3 )). r^. is differentiation of r once with respect to t.

$G i v e n t h e s p a c e c u r v e r = r (t), s h o w$ $t h a t i t s t o r s i o n τ i s g i v e n b y$ $τ = \frac{\dot{r} ∙ \overset{. .}{r} \times \overset{. . .}{r}}{∣ \dot{r} \times \overset{. .}{r} ∣^{2}} . I t m a y h e l p t o k n o w t h a t i t s$ $c u r v a t u r e i s n u m e r i c a l l y g i v e n b y κ = \frac{∣ \dot{r} \times \overset{. .}{r} ∣}{∣ \dot{r} ∣^{3}} .$ $\dot{r} i s d i f f e r e n t i a t i o n o f r o n c e w i t h$ $r e s p e c t t o t .$

Question Number 4772 Answers: 0 Comments: 4

Let z=Ax^2 +Bxy+Cy^2 . Find conditions on the constants A,B,C that ensure that the point (0,0,0) is a (i) local minimum, (ii) local maximum, (ii) saddle point.

$L e t z = {A x}^{2} + B x y + {C y}^{2} . F i n d c o n d i t i o n s$ $o n t h e c o n s t a n t s A, B, C t h a t e n s u r e$ $t h a t t h e p o i n t (0, 0, 0) i s a$ $(i) l o c a l m i n i m u m,$ $(i i) l o c a l m a x i m u m,$ $(i i) s a d d l e p o i n t .$

Question Number 4596 Answers: 0 Comments: 0

Use the definition of the limit of a function to prove that lim_((x,y)→(0,0)) ((x^4 +y^4 )/(x^2 +y^2 ))=0.

$U s e t h e d e f i n i t i o n o f t h e l i m i t o f a f u n c t i o n$ $t o p r o v e t h a t \lim_{(x, y) \to (0, 0)} \frac{x^{4} + y^{4}}{x^{2} + y^{2}} = 0 .$

Question Number 2045 Answers: 0 Comments: 2

Is the following series absolutely convergent? S_1 =Σ_(n=1) ^∞ (1/(n(n+1))) Is the following series absolutely convergent? S_2 =Σ_(n=1) ^∞ ((1/n)− (1/(n+1)))

$I s t h e f o l l o w i n g s e r i e s a b s o l u t e l y c o n v e r g e n t ?$ $S_{1} = \underset{n = 1}{\sum^{\infty}} \frac{1}{n (n + 1)}$ $I s t h e f o l l o w i n g s e r i e s a b s o l u t e l y c o n v e r g e n t ?$ $S_{2} = \underset{n = 1}{\sum^{\infty}} (\frac{1}{n} - \frac{1}{n + 1})$

Question Number 1865 Answers: 1 Comments: 0

proof that (√2) is an irrational number

$p r o o f t h a t \sqrt{2} i s a n i r r a t i o n a l n u m b e r$

Question Number 1537 Answers: 0 Comments: 0

Show by use of the characteristics of a vector−space, that the set x^→ ∈R_+ with the following operations ⊕ and builds a vector−space. ∗Use vector−space axioms ∗x^→ ⊕y^→ be x^→ ∙y^→ ∗for λ∈R set λ x^→ equal x^(→λ)

$S h o w b y u s e o f t h e c h a r a c t e r i s t i c s o f$ $a v e c t o r - s p a c e, t h a t t h e s e t \vec{x} \in R_{+}$ $with the following operations \oplus a n d$ $b u i l d s a v e c t o r - s p a c e .$ $* U s e v e c t o r - s p a c e a x i o m s$ $* \vec{x} \oplus \vec{y} b e \vec{x} \cdot \vec{y}$ $* f o r λ \in R s e t λ \vec{x} e q u a l \overset{\to λ}{x}$

Question Number 1479 Answers: 3 Comments: 0

Question Number 1016 Answers: 1 Comments: 0

[(x^→ ×a^→ )×b^→ ]×x^→ =c^→

$[(\vec{x} \times \vec{a}) \times \vec{b}] \times \vec{x} = \vec{c}$

Question Number 27 Answers: 1 Comments: 0

If f=x^2 zi−2y^3 z^2 j+xy^2 zk. Find div f, curl f, at(1, −1, 1).

$If f = x^{2} z i - 2 y^{3} z^{2} j + {x y}^{2} z k . Find d i v f, c u r l f,$ $a t (1, - 1, 1) .$

Question Number 4 Answers: 1 Comments: 0

If F(y(∂f/∂z)−z(∂f/∂y))i+(z(∂f/∂x)−x(∂f/∂z))j+(x(∂f/∂y)−y(∂f/∂x))k prove that F=r×▽f.

$If$ $F (y \frac{\partial f}{\partial z} - z \frac{\partial f}{\partial y}) i + (z \frac{\partial f}{\partial x} - x \frac{\partial f}{\partial z}) j + (x \frac{\partial f}{\partial y} - y \frac{\partial f}{\partial x}) k$ $prove that F = r \times ▽ f .$

Question Number 2 Answers: 1 Comments: 0

Find grad log ∣r∣.

$Find grad l o g ∣ r ∣ .$

Pg 1 Pg 2 Pg 3 Pg 4 Pg 5 Pg 6

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