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Determine the nature of the stationary value

X = t³ - 3t + ty²


Let r=xi^+yj^+zk^ and r=||r||. Show that:

∇(lnr)=r/r^2.

and


∇×((r^n)r)=0.


If A(u) is a differentiable vector function of u and ||A(u)||=1 , prove that dA/du is perpendicular to A .


Find the derivative of y= √x^3 using first principle


Let E:=Exi+Eyj+Ezk  and H:=Hxi+Hyj+Hzk be two vectors assumed to have continuous partial derivatives (of second order at least) with respect to position and time. Suppose further that E and H satisfy the equations:

∇⋅E=0,∇⋅H=0,∇×E=−1/c. ∂H/∂t, ∇×H=1/c. ∂E/∂t

prove that E and H satisfy the equation

∇^2Ei =1/c^2. ∂^2Ei/∂t^2  and ∇^2Hi=1/c^2. ∂^2Hi/∂t^2

Here, i=x,y  or z.

Hint: Use the fact that∇×(∇×V)=∇(∇⋅V)−∇^V.



The gain of an amplifier is found to be G = 20 log(10Vout) Determine equations for: dG dVout d2G dVout?


Differentiate of the following functions with respect to x:


i) 𝑡𝑎𝑛𝑥∙ 𝑙𝑛(𝑠𝑖𝑛𝑥)


ii) √𝑐𝑜𝑡√𝑥


iii) 𝑒 ln (𝑡𝑎𝑛5𝑥)


iv) 𝑆𝑖𝑛2


{ln(𝑠𝑒𝑐𝑥)}


v) ln (𝑐𝑜𝑠𝑒𝑐𝑥)/𝑥


If B(u)

 is a differentiable vector function of u

 and ||B(u)||=1

, prove that du

 is perpendicular to B



Let E:=E

x

i

^

+E

y

j

^

+E

z

k

^

 and H:=H

x

i

^

+H

y

j

^

+H

z

k

^

 be two vectors assumed to have continuous partial derivatives (of second order at least) with

respect to position and time. Suppose further that E

 and H

 satisfy the equations:


∇⋅E=0,∇⋅H=0,∇×E=−1

c


H

∂t


,∇×H=1

c


E

∂t


prove that E

 and H

 satisfy the equation


2

E

i

=1

c

2


2

E

i

∂t

2


 and ∇

2

H

i

=1

c

2


2

H

i

∂t

2


Here, i=x,y

 or z

.

Hint: Use the fact that∇×(∇×V)=∇(∇⋅V)−∇

2

V.





Evaluate: ∫4 𝑥𝑒𝑥 dx.