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Vector CalculusQuestion and Answers: Page 6

Question Number 5452 Answers: 1 Comments: 0

5×6

$$\mathrm{5}×\mathrm{6} \\ $$

Question Number 5153 Answers: 1 Comments: 0

proof e^(iΘ) =cos(Θ)+isin(Θ)

$${proof}\:\:\:\:{e}^{{i}\Theta} ={cos}\left(\Theta\right)+{isin}\left(\Theta\right) \\ $$

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.

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.

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

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.

$${Use}\:{the}\:{definition}\:{of}\:{the}\:{limit}\:{of}\:{a}\:{function} \\ $$$${to}\:{prove}\:{that}\:\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\frac{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} }{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\mathrm{0}. \\ $$

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)))

$${I}\mathrm{s}\:{the}\:{following}\:{series}\:{absolutely}\:{convergent}? \\ $$$${S}_{\mathrm{1}} =\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)} \\ $$$${Is}\:{the}\:{following}\:{series}\:{absolutely}\:{convergent}? \\ $$$${S}_{\mathrm{2}} =\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\frac{\mathrm{1}}{{n}}−\:\frac{\mathrm{1}}{{n}+\mathrm{1}}\right) \\ $$

Question Number 1865 Answers: 1 Comments: 0

proof that (√2) is an irrational number

$${proof}\:{that}\:\sqrt{\mathrm{2}}\:{is}\:{an}\:{irrational}\:{number} \\ $$$$ \\ $$

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)

$${Show}\:{by}\:{use}\:{of}\:{the}\:{characteristics}\:{of} \\ $$$${a}\:{vector}−{space},\:{that}\:{the}\:{set}\:\overset{\rightarrow} {{x}}\in\mathbb{R}_{+} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{following}\:\mathrm{operations}\:\oplus\:{and}\: \\ $$$${builds}\:{a}\:{vector}−{space}. \\ $$$$\ast{Use}\:{vector}−{space}\:{axioms} \\ $$$$\ast\overset{\rightarrow} {{x}}\oplus\overset{\rightarrow} {{y}}\:\:\:\:{be}\:\:\:\overset{\rightarrow} {{x}}\centerdot\overset{\rightarrow} {{y}} \\ $$$$\ast{for}\:\lambda\in\mathbb{R}\:\:{set}\:\:\lambda \overset{\rightarrow} {{x}}\:\:{equal}\:\overset{\rightarrow\lambda} {{x}} \\ $$

Question Number 1479 Answers: 3 Comments: 0

$$ \\ $$$$ \\ $$

Question Number 1016 Answers: 1 Comments: 0

[(x^→ ×a^→ )×b^→ ]×x^→ =c^→

$$\left[\left(\overset{\rightarrow} {{x}}×\overset{\rightarrow} {{a}}\right)×\overset{\rightarrow} {{b}}\right]×\overset{\rightarrow} {{x}}=\overset{\rightarrow} {{c}} \\ $$

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).

$$\mathrm{If}\:\boldsymbol{\mathrm{f}}={x}^{\mathrm{2}} {z}\boldsymbol{\mathrm{i}}−\mathrm{2}{y}^{\mathrm{3}} {z}^{\mathrm{2}} \boldsymbol{\mathrm{j}}+{xy}^{\mathrm{2}} {z}\boldsymbol{\mathrm{k}}.\:\mathrm{Find}\:{div}\:\boldsymbol{\mathrm{f}},\:{curl}\:\boldsymbol{\mathrm{f}},\: \\ $$$${at}\left(\mathrm{1},\:−\mathrm{1},\:\mathrm{1}\right). \\ $$

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.

$$\mathrm{If} \\ $$$$\mathrm{F}\left({y}\frac{\partial{f}}{\partial{z}}−{z}\frac{\partial{f}}{\partial{y}}\right)\boldsymbol{\mathrm{i}}+\left({z}\frac{\partial{f}}{\partial{x}}−{x}\frac{\partial{f}}{\partial{z}}\right)\boldsymbol{\mathrm{j}}+\left({x}\frac{\partial{f}}{\partial{y}}−{y}\frac{\partial{f}}{\partial{x}}\right)\boldsymbol{\mathrm{k}} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\:\:\boldsymbol{\mathrm{F}}=\boldsymbol{\mathrm{r}}×\bigtriangledown{f}. \\ $$

Question Number 2 Answers: 1 Comments: 0

Find grad log ∣r∣.

$$\mathrm{Find}\:\mathrm{grad}\:{log}\:\mid\boldsymbol{\mathrm{r}}\mid. \\ $$

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

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