Chapter 1: Q26P (page 24)
Calculate the Laplacian of the following functions:
(a)
(b)
(c) .
(d)
Short Answer
The value of
is 2.
The value of
is
.
The value of
is 0.
The value of
is
.
Step by step solution
Define the laplacian
The divergence of gradient is called as Laplacian of a vector.The vector is defined as
. the
operator is defined as
. The value of Laplacian
is obtained as
Here,is the second ordered partial derivative of function.
Compute
(a)
To compute an expression substitute the vectors and other required expression and then simplify
The vectoris defined as
. The Laplacian of the vector o
is defined as
.
Substitutefor
into
.
Thus the value of is 2.
Compute
(b)
To compute an expression substitute the vectors and other required expression and then simplify.
The vectoris defined as
. The Laplacian of the vector
is defined as
.
Substitutefor
into
.
Solve further as,
Thus, the value ofis
.
Compute
(c)
To compute an expression substitute the vectors and other required expression and then simplify.
The vectoris defined as
. The Laplacian of the vector
is defined as
.
Substitutefor
into
.
Solve further as,
Thus, the value ofis 0.
Compute
(d)
To compute an expression substitute the vectors and other required expression and then simplify.
The vectoris defined as
. The Laplacian of the vector
is defined as
.
Substitutefor
into
.
Solve further as,
Thus the value ofis
.
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