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.