Vector Analysis
BATANGAS STATE UNIVERSITY College of Engineering, Architecture, Fine Arts and Computing Sciences
ECE / ICE / MexE Department
ECE 352 VECTOR ANALYSIS
DEL OPERATOR
GROUP 3 Andaya, Rizalyn Ramos, Maria Issa P.
∇
Del is a symbol used in mathematics, in particular, in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇. Del may denote the gradient (locally steepest slope), the divergence of a vector field, or the curl (rotation) of a vector field. The symbol ∇ can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product of scalars, dot product, and cross product, respectively, of the del "operator" with the field.
A. GRADIENT If a scalar function , , is continuously differentiable with respect to its variables x, y, z through the region, then the gradient of , written grad , is defined as the vector grad = Using the vector differential operator ∇, ∇= We can write the gradient of as
+
+
+
+
grad = ∇ = ( = Example: If =
+ +
+ +
)
, determine the grad at the point P(1,3,2).
By definition: grad = ∇ =
+
+
All we have to do then is to find the partial derivatives at x=1, y=3, z=2 and insert their values.
Since = = =
= = = = 2(1) (3) (2)3 + (3)2(2)2 (1)2(2)3 + 2(1) (3) (2)2 3(1) (3) (2)2 + 2(1) (3)2(2) = 84 = 32 = 72
grad
= ∇ = 84 i + 32 j +72 k
B. DIVERGENCE If a vector function f= [f1, f2, f3], where f1, f2 and f3 are scalar functions, then its scalar or dot product with the symbolic vector ∇ is =[ ]
= Hence, a vector function f is transformed into a scalar function when operated on from the left by ∇. This scalar function is called the divergence of the vector function f and is written as div f: div = =
Example: Find the divergence of f=xyz i + x2y2z j + yz3 k Solution div = =
= Curl or Rot
= yz + 2 x 2yz + 3y z 2
-is a vector operator
ECE / ICE / MexE Department
ECE 352 VECTOR ANALYSIS
DEL OPERATOR
GROUP 3 Andaya, Rizalyn Ramos, Maria Issa P.
∇
Del is a symbol used in mathematics, in particular, in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇. Del may denote the gradient (locally steepest slope), the divergence of a vector field, or the curl (rotation) of a vector field. The symbol ∇ can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product of scalars, dot product, and cross product, respectively, of the del "operator" with the field.
A. GRADIENT If a scalar function , , is continuously differentiable with respect to its variables x, y, z through the region, then the gradient of , written grad , is defined as the vector grad = Using the vector differential operator ∇, ∇= We can write the gradient of as
+
+
+
+
grad = ∇ = ( = Example: If =
+ +
+ +
)
, determine the grad at the point P(1,3,2).
By definition: grad = ∇ =
+
+
All we have to do then is to find the partial derivatives at x=1, y=3, z=2 and insert their values.
Since = = =
= = = = 2(1) (3) (2)3 + (3)2(2)2 (1)2(2)3 + 2(1) (3) (2)2 3(1) (3) (2)2 + 2(1) (3)2(2) = 84 = 32 = 72
grad
= ∇ = 84 i + 32 j +72 k
B. DIVERGENCE If a vector function f= [f1, f2, f3], where f1, f2 and f3 are scalar functions, then its scalar or dot product with the symbolic vector ∇ is =[ ]
= Hence, a vector function f is transformed into a scalar function when operated on from the left by ∇. This scalar function is called the divergence of the vector function f and is written as div f: div = =
Example: Find the divergence of f=xyz i + x2y2z j + yz3 k Solution div = =
= Curl or Rot
= yz + 2 x 2yz + 3y z 2
-is a vector operator