Math252CalculusIII:IntroductiontoVectorFields by: javier
Math 252 Calculus III: Introduction to Vector Fields
by: javier
A Vector Field Definition & Examples
Definition & Examples
Definition: A function that assigns a vector to each point is called a vector field.
Definition & Examples
Definition: A function that assigns a vector to each point is called a vector field. typically F : R2 → R2
example f(x, y) = ⟨x + y, 3x2 y⟩
Definition & Examples
We have already seen a very famous vector field,
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it.
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it. Consider the surface function F(x, y) = 9 − x2 − y2
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it. Consider the surface function F(x, y) = 9 − x2 − y2
We saw that the gradient, ∇~F assigns to each point a vector, namely the vector pointing in the direction of fastest increase. Thus, ∇~F(x, y) = ⟨−2x, −2y⟩
is an example of a vector field.
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it. Consider the surface function F(x, y) = 9 − x2 − y2
We saw that the gradient, ∇~F assigns to each point a vector, namely the vector pointing in the direction of fastest increase. Thus, ∇~F(x, y) = ⟨−2x, −2y⟩
is an example of a vector field.
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it. Consider the surface function F(x, y) = 9 − x2 − y2
We saw that the gradient, ∇~F assigns to each point a vector, namely the vector pointing in the direction of fastest increase. Thus, ∇~F(x, y) = ⟨−2x, −2y⟩
is an example of a vector field.
Definition & Examples
We can generate such a vector field from any smooth differentiable surface.
Definition & Examples
We can generate such a vector field from any smooth differentiable surface. Consider the surface function F(x, y) = 3x2 + y2 − 5
Definition & Examples
We can generate such a vector field from any smooth differentiable surface. Consider the surface function F(x, y) = 3x2 + y2 − 5
Then ∇~F(x, y) = ⟨6x, 2y⟩
is an example of a vector field.
Definition & Examples
We can generate such a vector field from any smooth differentiable surface. Consider the surface function F(x, y) = 3x2 + y2 − 5
Then ∇~F(x, y) = ⟨6x, 2y⟩
is an example of a vector field.
Definition & Examples
We can generate such a vector field from any smooth differentiable surface. Consider the surface function F(x, y) = 3x2 + y2 − 5
Then ∇~F(x, y) = ⟨6x, 2y⟩
is an example of a vector field.
Definition & Examples
Another Example:
~F = ⟨x, 0⟩
Definition & Examples
Another Example:
~F = ⟨x, 0⟩
Definition & Examples
Another Example:
~F = ⟨0, x⟩
Definition & Examples
Another Example:
~F = ⟨0, x⟩
Definition & Examples
Another Example:
~F = ⟨−y, x⟩
Definition & Examples
Another Example:
~F = ⟨−y, x⟩
A Vector Field Famous Operators on Vector Fields
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨3x, 4y⟩ div (~F) = ∇ · ~F
div (~F) =
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨3x, 4y⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is expanding or collapsing at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨3x, 4y⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is expanding or collapsing at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨1, 2⟩ div (~F) = ∇ · ~F
div (~F) =
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨1, 2⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨1, 2⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨−x, −y⟩ div (~F) = ∇ · ~F
div (~F) =
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨−x, −y⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨−x, −y⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨cos(πx/4), sin(πx/4)⟩ div (~F) = ∇ · ~F div (~F) =
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨cos(πx/4), sin(πx/4)⟩ div (~F) = ∇ · ~F div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨cos(πx/4), sin(πx/4)⟩ div (~F) = ∇ · ~F div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨−y, x⟩ curl (~F) = ∇ × ~F
curl (~F) =
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨−y, x⟩ curl (~F) = ∇ × ~F
curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨−y, x⟩ curl (~F) = ∇ × ~F
curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨3, −2⟩ curl (~F) = ∇ × ~F
curl (~F) =
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨3, −2⟩ curl (~F) = ∇ × ~F
curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨3, −2⟩ curl (~F) = ∇ × ~F
curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: ( given )
~F = ⟨−y cos
1 6
πy , x cos
(1
) πx ⟩ 6
curl (~F) = ∇ × ~F curl (~F) =
Famous Operators on Vector Fields
CURL Compute the curl: ( given )
~F = ⟨−y cos
1 6
πy , x cos
(1
) πx ⟩ 6
curl (~F) = ∇ × ~F curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: ( given )
~F = ⟨−y cos
1 6
πy , x cos
(1
) πx ⟩ 6
curl (~F) = ∇ × ~F curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL
A Vector Field VOCABULARY & famous fields
VOCABULARY & famous fields
VOCABULARY If a vector field, F, comes from the gradient of some function, f, then F is called a gradient vector field , the function f is called its potential , and a bunch of nice things follow.
VOCABULARY & famous fields Assuming a the function is smooth, continuous and differentiable on a simply connected region, then QUICK CHECK for ”gradient-ness” (in two dimensions) Suppose F(x, y) = ⟨M, N⟩
VOCABULARY & famous fields Assuming a the function is smooth, continuous and differentiable on a simply connected region, then QUICK CHECK for ”gradient-ness” (in two dimensions) Suppose F(x, y) = ⟨M, N⟩
then F is a gradient vector field iff My = Nx
VOCABULARY & famous fields Assuming a the function is smooth, continuous and differentiable on a simply connected region, then QUICK CHECK for ”gradient-ness” (in two dimensions) Suppose F(x, y) = ⟨M, N⟩
then F is a gradient vector field iff My = Nx
(in three dimensions) Suppose F(x, y, z) = ⟨M, N, P⟩
VOCABULARY & famous fields Assuming a the function is smooth, continuous and differentiable on a simply connected region, then QUICK CHECK for ”gradient-ness” (in two dimensions) Suppose F(x, y) = ⟨M, N⟩
then F is a gradient vector field iff My = Nx
(in three dimensions) Suppose F(x, y, z) = ⟨M, N, P⟩
then F is a gradient vector field iff ∇×F=0
NEED: nice F, continous differentaialbe over a simply connected domain.
by: javier
A Vector Field Definition & Examples
Definition & Examples
Definition: A function that assigns a vector to each point is called a vector field.
Definition & Examples
Definition: A function that assigns a vector to each point is called a vector field. typically F : R2 → R2
example f(x, y) = ⟨x + y, 3x2 y⟩
Definition & Examples
We have already seen a very famous vector field,
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it.
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it. Consider the surface function F(x, y) = 9 − x2 − y2
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it. Consider the surface function F(x, y) = 9 − x2 − y2
We saw that the gradient, ∇~F assigns to each point a vector, namely the vector pointing in the direction of fastest increase. Thus, ∇~F(x, y) = ⟨−2x, −2y⟩
is an example of a vector field.
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it. Consider the surface function F(x, y) = 9 − x2 − y2
We saw that the gradient, ∇~F assigns to each point a vector, namely the vector pointing in the direction of fastest increase. Thus, ∇~F(x, y) = ⟨−2x, −2y⟩
is an example of a vector field.
Definition & Examples
We have already seen a very famous vector field, in fact we have seen many, as every surface function has one associated with it. Consider the surface function F(x, y) = 9 − x2 − y2
We saw that the gradient, ∇~F assigns to each point a vector, namely the vector pointing in the direction of fastest increase. Thus, ∇~F(x, y) = ⟨−2x, −2y⟩
is an example of a vector field.
Definition & Examples
We can generate such a vector field from any smooth differentiable surface.
Definition & Examples
We can generate such a vector field from any smooth differentiable surface. Consider the surface function F(x, y) = 3x2 + y2 − 5
Definition & Examples
We can generate such a vector field from any smooth differentiable surface. Consider the surface function F(x, y) = 3x2 + y2 − 5
Then ∇~F(x, y) = ⟨6x, 2y⟩
is an example of a vector field.
Definition & Examples
We can generate such a vector field from any smooth differentiable surface. Consider the surface function F(x, y) = 3x2 + y2 − 5
Then ∇~F(x, y) = ⟨6x, 2y⟩
is an example of a vector field.
Definition & Examples
We can generate such a vector field from any smooth differentiable surface. Consider the surface function F(x, y) = 3x2 + y2 − 5
Then ∇~F(x, y) = ⟨6x, 2y⟩
is an example of a vector field.
Definition & Examples
Another Example:
~F = ⟨x, 0⟩
Definition & Examples
Another Example:
~F = ⟨x, 0⟩
Definition & Examples
Another Example:
~F = ⟨0, x⟩
Definition & Examples
Another Example:
~F = ⟨0, x⟩
Definition & Examples
Another Example:
~F = ⟨−y, x⟩
Definition & Examples
Another Example:
~F = ⟨−y, x⟩
A Vector Field Famous Operators on Vector Fields
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨3x, 4y⟩ div (~F) = ∇ · ~F
div (~F) =
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨3x, 4y⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is expanding or collapsing at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨3x, 4y⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is expanding or collapsing at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨1, 2⟩ div (~F) = ∇ · ~F
div (~F) =
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨1, 2⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨1, 2⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨−x, −y⟩ div (~F) = ∇ · ~F
div (~F) =
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨−x, −y⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨−x, −y⟩ div (~F) = ∇ · ~F
div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨cos(πx/4), sin(πx/4)⟩ div (~F) = ∇ · ~F div (~F) =
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨cos(πx/4), sin(πx/4)⟩ div (~F) = ∇ · ~F div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
DIVERGENCE Compute the divergence: given ~F = ⟨cos(πx/4), sin(πx/4)⟩ div (~F) = ∇ · ~F div (~F) =
divergence describes how much a field is exploding or imploding at any given point.
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨−y, x⟩ curl (~F) = ∇ × ~F
curl (~F) =
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨−y, x⟩ curl (~F) = ∇ × ~F
curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨−y, x⟩ curl (~F) = ∇ × ~F
curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨3, −2⟩ curl (~F) = ∇ × ~F
curl (~F) =
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨3, −2⟩ curl (~F) = ∇ × ~F
curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: given ~F = ⟨3, −2⟩ curl (~F) = ∇ × ~F
curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: ( given )
~F = ⟨−y cos
1 6
πy , x cos
(1
) πx ⟩ 6
curl (~F) = ∇ × ~F curl (~F) =
Famous Operators on Vector Fields
CURL Compute the curl: ( given )
~F = ⟨−y cos
1 6
πy , x cos
(1
) πx ⟩ 6
curl (~F) = ∇ × ~F curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL Compute the curl: ( given )
~F = ⟨−y cos
1 6
πy , x cos
(1
) πx ⟩ 6
curl (~F) = ∇ × ~F curl (~F) =
the curl describes how much a field is curling, ie how much a floating cork would rotate.
Famous Operators on Vector Fields
CURL
A Vector Field VOCABULARY & famous fields
VOCABULARY & famous fields
VOCABULARY If a vector field, F, comes from the gradient of some function, f, then F is called a gradient vector field , the function f is called its potential , and a bunch of nice things follow.
VOCABULARY & famous fields Assuming a the function is smooth, continuous and differentiable on a simply connected region, then QUICK CHECK for ”gradient-ness” (in two dimensions) Suppose F(x, y) = ⟨M, N⟩
VOCABULARY & famous fields Assuming a the function is smooth, continuous and differentiable on a simply connected region, then QUICK CHECK for ”gradient-ness” (in two dimensions) Suppose F(x, y) = ⟨M, N⟩
then F is a gradient vector field iff My = Nx
VOCABULARY & famous fields Assuming a the function is smooth, continuous and differentiable on a simply connected region, then QUICK CHECK for ”gradient-ness” (in two dimensions) Suppose F(x, y) = ⟨M, N⟩
then F is a gradient vector field iff My = Nx
(in three dimensions) Suppose F(x, y, z) = ⟨M, N, P⟩
VOCABULARY & famous fields Assuming a the function is smooth, continuous and differentiable on a simply connected region, then QUICK CHECK for ”gradient-ness” (in two dimensions) Suppose F(x, y) = ⟨M, N⟩
then F is a gradient vector field iff My = Nx
(in three dimensions) Suppose F(x, y, z) = ⟨M, N, P⟩
then F is a gradient vector field iff ∇×F=0
NEED: nice F, continous differentaialbe over a simply connected domain.
