The Central Idea
Suppose a scalar field assigns one value to every point:
\[ f(x,y,z). \]We may want more than the value itself. We may want to know:
- which direction makes \(f\) increase fastest;
- how rapidly \(f\) can increase;
- which directions leave \(f\) unchanged to first order;
- how the field changes near a point.
The gradient packages the relevant partial derivatives into a vector:
\[ \boxed{\nabla f}. \]The Gradient Operator
The symbol
\[ \nabla \]is called nabla or del.
In Cartesian coordinates,
\[ \nabla = \left\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right\rangle. \]It is a differential operator. When applied to a scalar field, it produces the gradient vector field.
Gradient in Two Dimensions
If
\[ f=f(x,y), \]then
\[ \boxed{ \nabla f(x,y) = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle }. \]Using subscript notation,
\[ \nabla f = \langle f_x,f_y\rangle. \]Gradient in Three Dimensions
If
\[ f=f(x,y,z), \]then
\[ \boxed{ \nabla f(x,y,z) = \left\langle f_x,f_y,f_z \right\rangle }. \]Each component measures the rate of change of the scalar field along one coordinate direction.
A Gradient Is a Vector Field
The original object
\[ f(x,y) \]is scalar-valued.
Its gradient
\[ \nabla f(x,y) \]is vector-valued.
This change of mathematical type is fundamental:
\[ \boxed{\text{scalar field} \xrightarrow{\ \nabla\ } \text{vector field}}. \]Example: Computing a Two-Dimensional Gradient
Let
\[ f(x,y)=x^2+3xy+2y^2. \]Differentiate with respect to \(x\):
\[ f_x=2x+3y. \]Differentiate with respect to \(y\):
\[ f_y=3x+4y. \]Therefore,
\[ \boxed{ \nabla f(x,y) = \langle2x+3y,\;3x+4y\rangle }. \]Evaluating a Gradient at a Point
For
\[ \nabla f(x,y) = \langle2x+3y,\;3x+4y\rangle, \]evaluate at \((1,2)\):
\[ \nabla f(1,2) = \langle2(1)+3(2),\;3(1)+4(2)\rangle. \]Hence,
\[ \boxed{\nabla f(1,2)=\langle8,11\rangle}. \]The scalar field has one gradient vector at that particular point.
Example: Computing a Three-Dimensional Gradient
Let
\[ f(x,y,z)=x^2y+yz^3+4z. \]Then
\[ f_x=2xy, \] \[ f_y=x^2+z^3, \]and
\[ f_z=3yz^2+4. \]Therefore,
\[ \boxed{ \nabla f = \langle 2xy,\; x^2+z^3,\; 3yz^2+4 \rangle }. \]What the Gradient Means
The gradient is not merely a convenient list of partial derivatives.
At a point where
\[ \nabla f\ne\mathbf{0}, \]the gradient points in the direction of the greatest instantaneous increase of the scalar field.
Its magnitude gives the greatest directional rate of change available at that point.
Direction of Steepest Increase
Suppose
\[ \nabla f(a,b)=\langle3,4\rangle. \]Then the field increases most rapidly in the direction of
\[ \langle3,4\rangle. \]The corresponding unit direction is
\[ \frac{\langle3,4\rangle}{\sqrt{3^2+4^2}} = \boxed{\left\langle\frac35,\frac45\right\rangle}. \]Direction of Steepest Decrease
If the gradient gives steepest increase, then the opposite direction gives steepest decrease.
Thus,
\[ -\nabla f \]points in the direction of fastest decrease.
This idea is the geometric foundation of gradient descent in optimization.
Maximum Rate of Increase
The maximum rate at which \(f\) can increase per unit distance is
\[ \boxed{\|\nabla f\|}. \]If
\[ \nabla f(a,b)=\langle3,4\rangle, \]then
\[ \|\nabla f(a,b)\| = \sqrt{3^2+4^2} = \boxed{5}. \]Directional Derivatives Revisited
A directional derivative measures the rate of change of a scalar field in a chosen direction.
If \(\mathbf{u}\) is a unit vector, then
\[ \boxed{ D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u} }. \]This formula connects directional derivatives directly to the gradient.
Why the Direction Vector Must Be a Unit Vector
The directional derivative is intended to measure change per unit distance.
If a direction is given by a nonzero vector \(\mathbf{v}\), first normalize it:
\[ \boxed{ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} }. \]Then compute
\[ D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u}. \]Example: Directional Derivative from the Gradient
Suppose
\[ \nabla f(1,2)=\langle8,11\rangle \]and we want the directional derivative in the direction
\[ \mathbf{v}=\langle3,4\rangle. \]Normalize:
\[ \mathbf{u} = \left\langle\frac35,\frac45\right\rangle. \]Then
\[ D_{\mathbf{u}}f = \langle8,11\rangle \cdot \left\langle\frac35,\frac45\right\rangle. \]Therefore,
\[ D_{\mathbf{u}}f = \frac{24}{5}+\frac{44}{5} = \boxed{\frac{68}{5}}. \]The Dot-Product Interpretation
Because
\[ \nabla f\cdot\mathbf{u} = \|\nabla f\|\|\mathbf{u}\|\cos\theta, \]and \(\|\mathbf{u}\|=1\),
\[ \boxed{ D_{\mathbf{u}}f = \|\nabla f\|\cos\theta }. \]Here \(\theta\) is the angle between the gradient and the chosen direction.
Why the Gradient Gives the Maximum
Since
\[ -1\le\cos\theta\le1, \]the largest possible value of
\[ \|\nabla f\|\cos\theta \]occurs when
\[ \theta=0. \]That means the chosen direction is the same as the gradient direction.
Therefore,
\[ \boxed{ \max_{\|\mathbf{u}\|=1}D_{\mathbf{u}}f = \|\nabla f\| }. \]Why the Negative Gradient Gives the Minimum
The smallest directional derivative occurs when
\[ \theta=\pi. \]Then
\[ \cos\pi=-1, \]so
\[ \boxed{ \min_{\|\mathbf{u}\|=1}D_{\mathbf{u}}f = -\|\nabla f\| }. \]Directions of No First-Order Change
If a direction \(\mathbf{u}\) is perpendicular to the gradient, then
\[ \nabla f\cdot\mathbf{u}=0. \]Therefore,
\[ \boxed{D_{\mathbf{u}}f=0}. \]Moving in such a direction produces no first-order change in the scalar field.
Level Curves and the Gradient
A level curve is defined by
\[ f(x,y)=c. \]Along that curve, the value of \(f\) stays constant.
Therefore, motion tangent to the level curve produces zero directional derivative.
Since the gradient is perpendicular to every tangent direction,
\[ \boxed{ \nabla f \text{ is normal to the level curve} }. \]Example: Gradient Normal to a Circle
Let
\[ f(x,y)=x^2+y^2. \]Then
\[ \nabla f = \langle2x,2y\rangle. \]The level curves
\[ x^2+y^2=c \]are circles centered at the origin.
The gradient \(\langle2x,2y\rangle\) points radially outward, exactly normal to each circle.
Level Surfaces and the Gradient
In three dimensions, a level surface is defined by
\[ f(x,y,z)=c. \]At a regular point where
\[ \nabla f\ne\mathbf{0}, \]the gradient is normal to the level surface.
This gives an efficient way to find normal vectors and tangent planes.
Example: Normal Vector to a Sphere
Consider the sphere
\[ x^2+y^2+z^2=25. \]Define
\[ f(x,y,z)=x^2+y^2+z^2. \]Then
\[ \nabla f = \langle2x,2y,2z\rangle. \]At \((3,4,0)\),
\[ \nabla f(3,4,0) = \boxed{\langle6,8,0\rangle}. \]This vector is normal to the sphere at that point.
Tangent Plane from a Gradient
If the level surface
\[ f(x,y,z)=c \]passes through \((x_0,y_0,z_0)\), then a normal vector is
\[ \nabla f(x_0,y_0,z_0). \]Therefore, the tangent plane can be written
\[ \boxed{ \nabla f(x_0,y_0,z_0) \cdot \langle x-x_0,\; y-y_0,\; z-z_0 \rangle =0 }. \]Gradient Fields
Any vector field of the form
\[ \boxed{\mathbf{F}=\nabla f} \]is called a gradient field.
The scalar field \(f\) is often called a potential function for \(\mathbf{F}\).
Example: Building a Gradient Field
Let
\[ f(x,y)=x^2y+y^3. \]Then
\[ f_x=2xy \]and
\[ f_y=x^2+3y^2. \]Therefore, the gradient field is
\[ \boxed{ \mathbf{F}(x,y) = \nabla f = \langle2xy,\;x^2+3y^2\rangle }. \]Potential Functions
If
\[ \mathbf{F} = \nabla f, \]then \(f\) is a scalar potential for the vector field.
Different potential functions can generate the same gradient field if they differ only by a constant:
\[ \nabla(f+C)=\nabla f. \]Therefore, potentials are generally determined only up to an additive constant.
Recovering a Potential Function
Suppose
\[ \mathbf{F}(x,y) = \langle P(x,y),Q(x,y)\rangle \]and we seek \(f\) such that
\[ \nabla f=\mathbf{F}. \]Then
\[ f_x=P \]and
\[ f_y=Q. \]We can integrate one component and use the other component to determine the remaining function of the other variable.
Example: Recovering a Potential
Let
\[ \mathbf{F}(x,y) = \langle2xy,\;x^2+3y^2\rangle. \]From
\[ f_x=2xy, \]integrate with respect to \(x\):
\[ f(x,y)=x^2y+g(y). \]Differentiate with respect to \(y\):
\[ f_y=x^2+g'(y). \]Match this with
\[ x^2+3y^2. \]Therefore,
\[ g'(y)=3y^2, \]so
\[ g(y)=y^3+C. \]A potential is
\[ \boxed{f(x,y)=x^2y+y^3+C}. \]Why the Extra Function Appears
When integrating
\[ f_x=P(x,y) \]with respect to \(x\), any function depending only on \(y\) behaves like a constant with respect to \(x\).
Therefore, we write
\[ f(x,y) = \int P(x,y)\,dx+g(y), \]not merely a single constant.
Three-Dimensional Potential Functions
If
\[ \mathbf{F} = \langle P,Q,R\rangle \]is a gradient field, then
\[ f_x=P,\qquad f_y=Q,\qquad f_z=R. \]Recovering \(f\) requires matching all three component equations.
The same integration logic applies, but bookkeeping becomes more important.
Gradient Fields and Conservative Fields
A vector field that can be expressed as the gradient of a potential function is called conservative on the relevant domain.
\[ \boxed{ \mathbf{F}=\nabla f \quad\Longrightarrow\quad \mathbf{F}\text{ is a gradient field} }. \]The later lesson on conservative fields and potentials will develop the consequences carefully.
Mixed Partial Derivatives and a Useful Test
Suppose
\[ \mathbf{F}(x,y)=\langle P,Q\rangle \]and
\[ P=f_x,\qquad Q=f_y. \]If the relevant second partial derivatives are continuous, then
\[ P_y=f_{xy} \]and
\[ Q_x=f_{yx}. \]Therefore,
\[ \boxed{P_y=Q_x}. \]This is a useful necessary condition for a smooth two-dimensional gradient field and, on suitable domains, becomes part of a practical conservativeness test.
Example: Checking the Mixed Partials
Consider
\[ \mathbf{F}(x,y) = \langle2xy,\;x^2+3y^2\rangle. \]Here
\[ P_y=2x \]and
\[ Q_x=2x. \]The equality is consistent with the fact that this field is generated by the potential
\[ f=x^2y+y^3. \]A Field That Fails the Mixed-Partial Test
Consider
\[ \mathbf{F}(x,y) = \langle-y,x\rangle. \]Then
\[ P_y=-1 \]while
\[ Q_x=1. \]Since
\[ P_y\ne Q_x, \]this field cannot be the gradient of a twice continuously differentiable scalar potential on an open region containing the point under consideration.
Gradient and the Multivariable Chain Rule
Suppose a path is parameterized by
\[ \mathbf{r}(t) = \langle x(t),y(t),z(t)\rangle. \]If a scalar field \(f\) is evaluated along the path, then
\[ f(\mathbf{r}(t)) = f(x(t),y(t),z(t)). \]The multivariable chain rule gives
\[ \boxed{ \frac{d}{dt}f(\mathbf{r}(t)) = \nabla f(\mathbf{r}(t)) \cdot \mathbf{r}'(t) }. \]Interpretation Along a Path
In
\[ \frac{d}{dt}f(\mathbf{r}(t)) = \nabla f\cdot\mathbf{r}'(t), \]the vector
\[ \mathbf{r}'(t) \]describes how position changes, while
\[ \nabla f \]describes how the scalar field changes with position.
Their dot product gives the actual rate of change experienced along the path.
Example: Temperature Along a Moving Path
Suppose
\[ T(x,y)=x^2+y^2 \]and a particle moves along
\[ \mathbf{r}(t)=\langle t,2t\rangle. \]The gradient is
\[ \nabla T=\langle2x,2y\rangle. \]Along the path,
\[ \nabla T(\mathbf{r}(t)) = \langle2t,4t\rangle. \]Also,
\[ \mathbf{r}'(t)=\langle1,2\rangle. \]Therefore,
\[ \frac{dT}{dt} = \langle2t,4t\rangle\cdot\langle1,2\rangle = 2t+8t = \boxed{10t}. \]Critical Points of a Scalar Field
A differentiable scalar field has a critical point where
\[ \boxed{\nabla f=\mathbf{0}}. \]In two dimensions, this means solving
\[ f_x=0 \]and
\[ f_y=0 \]simultaneously.
Example: Finding a Critical Point
Let
\[ f(x,y)=x^2+4y^2-6x+8y. \]Then
\[ \nabla f = \langle2x-6,\;8y+8\rangle. \]Set both components equal to zero:
\[ 2x-6=0, \] \[ 8y+8=0. \]Thus,
\[ \boxed{(x,y)=(3,-1)}. \]What Happens When the Gradient Is Zero?
If
\[ \nabla f=\mathbf{0}, \]then every directional derivative computed from
\[ D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u} \]is zero.
This means there is no first-order direction of increase or decrease at that point.
The point may be a local maximum, local minimum, saddle point, or another critical configuration.
Gradient Descent
Optimization algorithms often move opposite the gradient:
\[ \boxed{ \mathbf{x}_{n+1} = \mathbf{x}_n - \alpha\nabla f(\mathbf{x}_n) }. \]Here \(\alpha>0\) is a step size.
The logic is simple: because \(\nabla f\) points uphill, \(-\nabla f\) points downhill.
Gradient Ascent
To increase a function rather than decrease it, one may move with the gradient:
\[ \boxed{ \mathbf{x}_{n+1} = \mathbf{x}_n + \alpha\nabla f(\mathbf{x}_n) }. \]Gradient ascent and gradient descent are basic local optimization methods.
Units of a Gradient
The units of a gradient are the units of the scalar field divided by units of distance.
If temperature is measured in degrees Celsius and position in meters, then
\[ \nabla T \]has units of degrees Celsius per meter.
Its magnitude measures the steepest spatial temperature change per meter.
Gradient of a Linear Scalar Field
Let
\[ f(x,y)=ax+by+c. \]Then
\[ f_x=a \]and
\[ f_y=b. \]Therefore,
\[ \boxed{\nabla f=\langle a,b\rangle}. \]The gradient is constant everywhere.
Gradient of a Radial Scalar Field
Consider
\[ f(x,y)=x^2+y^2. \]Then
\[ \nabla f = \langle2x,2y\rangle. \]This is an outward radial vector field.
The farther we move from the origin, the larger the gradient magnitude:
\[ \|\nabla f\| = 2\sqrt{x^2+y^2}. \]Gradient of a Gaussian-Type Field
Let
\[ f(x,y)=e^{-(x^2+y^2)}. \]Then
\[ f_x=-2xe^{-(x^2+y^2)} \]and
\[ f_y=-2ye^{-(x^2+y^2)}. \]Therefore,
\[ \boxed{ \nabla f = -2e^{-(x^2+y^2)} \langle x,y\rangle }. \]Away from the origin, the gradient points inward because the scalar field decreases as distance from the origin increases.
Gradient and Local Linear Approximation
Near a point \(\mathbf{a}\), a differentiable scalar field satisfies the first-order approximation
\[ \boxed{ f(\mathbf{a}+\Delta\mathbf{x}) \approx f(\mathbf{a}) + \nabla f(\mathbf{a}) \cdot \Delta\mathbf{x} }. \]The gradient therefore contains the coefficients of the best local linear model.
The Gradient Collects Sensitivities
Each partial derivative measures sensitivity to one coordinate:
\[ f_x,\qquad f_y,\qquad f_z. \]The gradient collects these coordinate sensitivities into one geometric object:
\[ \nabla f = \langle f_x,f_y,f_z\rangle. \]This is why gradients are useful in modeling systems with several inputs.
Gradient Fields in Physical Models
Gradients occur naturally when a scalar quantity varies through space.
- temperature gradients describe spatial thermal change;
- pressure gradients help drive fluid acceleration;
- concentration gradients describe spatial chemical variation;
- potential gradients are connected to force and field models;
- elevation gradients describe steepest uphill direction on terrain.
Sign Conventions in Physics
Some physical vector fields are defined as the negative gradient of a scalar potential:
\[ \boxed{\mathbf{F}=-\nabla V}. \]The negative sign means the vector field points toward decreasing potential.
The exact interpretation depends on the physical model and its conventions.
Gradient and Topographic Maps
Suppose
\[ h(x,y) \]gives terrain elevation.
Contour lines are level curves
\[ h(x,y)=c. \]The gradient
\[ \nabla h \]points uphill and perpendicular to the local contour line.
Closely spaced contours usually indicate a larger gradient magnitude than widely spaced contours.
Gradient and Normalized Direction
When
\[ \nabla f\ne\mathbf{0}, \]the unit vector in the direction of steepest increase is
\[ \boxed{ \frac{\nabla f}{\|\nabla f\|} }. \]This separates the gradient's directional information from its magnitude.
Common Error: Forgetting a Partial Derivative
For
\[ f(x,y,z), \]the gradient requires one component for each coordinate:
\[ \boxed{ \nabla f = \langle f_x,f_y,f_z\rangle }. \]Omitting a component changes the mathematical object.
Common Error: Treating the Gradient as a Scalar
The gradient itself is a vector:
\[ \nabla f. \]Its magnitude is a scalar:
\[ \|\nabla f\|. \]These objects answer different questions.
Common Error: Using a Nonunit Direction
If the requested direction is
\[ \mathbf{v}=\langle a,b\rangle, \]do not automatically compute
\[ \nabla f\cdot\mathbf{v}. \]First form
\[ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}. \]Then use
\[ D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u}. \]Common Error: Confusing a Level Curve with a Gradient Curve
A level curve satisfies
\[ f(x,y)=c. \]The gradient is generally perpendicular to that curve, not tangent to it.
The gradient field and the level-curve family encode different but closely related information.
Common Error: Assuming Every Vector Field Is a Gradient Field
A vector field
\[ \mathbf{F} \]is a gradient field only if there exists a scalar potential \(f\) such that
\[ \mathbf{F}=\nabla f. \]Many vector fields do not have such a potential on a given domain.
Common Error: Dropping the Extra Function During Integration
If
\[ f_x=P(x,y), \]then integration with respect to \(x\) gives
\[ f(x,y)=\int P(x,y)\,dx+g(y). \]The unknown function \(g(y)\) must be retained until the other component equation determines it.
A Reliable Procedure for Computing a Gradient
- Identify the scalar field and its variables.
- Compute the partial derivative with respect to each spatial variable.
- Place the partial derivatives in coordinate order.
- Write the result as a vector field.
- Evaluate at a point only after the general gradient is correct.
- Check units and domain restrictions when the problem is applied.
A Reliable Procedure for Directional Derivatives
- Compute \(\nabla f\).
- Evaluate \(\nabla f\) at the requested point.
- Normalize the given direction vector if necessary.
- Compute the dot product \(\nabla f\cdot\mathbf{u}\).
- Interpret the sign and magnitude of the result.
A Reliable Procedure for Finding a Potential
- Write \(\mathbf{F}=\langle P,Q\rangle\) or \(\langle P,Q,R\rangle\).
- Integrate one component with respect to its corresponding variable.
- Add an unknown function of the remaining variables.
- Differentiate the candidate potential using another variable.
- Match the result with the corresponding field component.
- Determine the unknown function.
- Use any remaining component as a consistency check.
- Include the final additive constant \(C\).
Example: Full Gradient Workflow
Let
\[ f(x,y)=x^3-2xy+y^2. \]First compute the partial derivatives:
\[ f_x=3x^2-2y, \] \[ f_y=-2x+2y. \]Therefore,
\[ \nabla f = \langle3x^2-2y,\;-2x+2y\rangle. \]At \((1,3)\),
\[ \nabla f(1,3) = \langle3-6,\;-2+6\rangle = \boxed{\langle-3,4\rangle}. \]The maximum rate of increase there is
\[ \|\nabla f(1,3)\| = \sqrt{(-3)^2+4^2} = \boxed{5}. \]The unit direction of steepest increase is
\[ \boxed{ \left\langle-\frac35,\frac45\right\rangle }. \]Example: Full Potential Workflow
Suppose
\[ \mathbf{F}(x,y) = \langle3x^2+2y,\;2x+4y\rangle. \]We seek \(f\) satisfying
\[ f_x=3x^2+2y. \]Integrate with respect to \(x\):
\[ f=x^3+2xy+g(y). \]Differentiate with respect to \(y\):
\[ f_y=2x+g'(y). \]Match this with
\[ 2x+4y. \]Then
\[ g'(y)=4y, \]so
\[ g(y)=2y^2+C. \]Therefore,
\[ \boxed{ f(x,y)=x^3+2xy+2y^2+C }. \]Practice 1: Compute a Gradient
Find \(\nabla f\) for
\[ f(x,y)=4x^2-3xy+y^2. \]Practice 2: Evaluate a Gradient
Using Practice 1, find
\[ \nabla f(2,-1). \]Practice 3: Maximum Rate of Increase
Using Practice 2, find the maximum rate of increase of \(f\) at \((2,-1)\).
Practice 4: Direction of Steepest Increase
Using Practice 2, find a unit vector pointing in the direction of steepest increase.
Practice 5: Directional Derivative
Let
\[ f(x,y)=x^2+y^2. \]Find the directional derivative at \((1,2)\) in the direction of
\[ \mathbf{v}=\langle3,4\rangle. \]Practice 6: Normal to a Level Curve
Let
\[ f(x,y)=x^2+4y^2. \]Find a normal vector to the level curve through \((2,1)\).
Practice 7: Normal to a Level Surface
For
\[ f(x,y,z)=x^2+y^2+z^2, \]find a normal vector to the level surface through \((1,2,2)\).
Practice 8: Potential Function
Find a potential function for
\[ \mathbf{F}(x,y) = \langle2x+y,\;x+6y\rangle. \]Practice 9: Mixed-Partial Check
Determine whether the mixed-partial condition is satisfied for
\[ \mathbf{F}(x,y) = \langle y^2,\;2xy+3y^2\rangle. \]Practice 10: Critical Point
Find the critical point of
\[ f(x,y)=x^2+y^2-4x+6y+10. \]Practice Answers
Practice 1
\[ f_x=8x-3y, \] \[ f_y=-3x+2y. \]Therefore,
\[ \boxed{ \nabla f = \langle8x-3y,\;-3x+2y\rangle }. \]Practice 2
\[ \nabla f(2,-1) = \langle16+3,\;-6-2\rangle = \boxed{\langle19,-8\rangle}. \]Practice 3
\[ \|\nabla f(2,-1)\| = \sqrt{19^2+(-8)^2} = \boxed{\sqrt{425}} = \boxed{5\sqrt{17}}. \]Practice 4
\[ \boxed{ \frac{\nabla f}{\|\nabla f\|} = \left\langle \frac{19}{5\sqrt{17}}, -\frac{8}{5\sqrt{17}} \right\rangle }. \]Practice 5
First,
\[ \nabla f=\langle2x,2y\rangle. \]At \((1,2)\),
\[ \nabla f(1,2)=\langle2,4\rangle. \]Normalize the direction:
\[ \mathbf{u} = \left\langle\frac35,\frac45\right\rangle. \]Then
\[ D_{\mathbf{u}}f = \langle2,4\rangle \cdot \left\langle\frac35,\frac45\right\rangle = \frac65+\frac{16}{5} = \boxed{\frac{22}{5}}. \]Practice 6
\[ \nabla f = \langle2x,8y\rangle. \]At \((2,1)\),
\[ \boxed{\nabla f(2,1)=\langle4,8\rangle}. \]Any nonzero scalar multiple of this vector is also a normal vector.
Practice 7
\[ \nabla f = \langle2x,2y,2z\rangle. \]At \((1,2,2)\),
\[ \boxed{\langle2,4,4\rangle}. \]Practice 8
From
\[ f_x=2x+y, \]integrate:
\[ f=x^2+xy+g(y). \]Then
\[ f_y=x+g'(y). \]Matching \(x+6y\) gives
\[ g'(y)=6y, \]so
\[ g(y)=3y^2+C. \]Therefore,
\[ \boxed{ f(x,y)=x^2+xy+3y^2+C }. \]Practice 9
Here
\[ P=y^2, \qquad Q=2xy+3y^2. \]Then
\[ P_y=2y \]and
\[ Q_x=2y. \]Therefore,
\[ \boxed{P_y=Q_x}. \]The mixed-partial condition is satisfied.
Practice 10
\[ \nabla f = \langle2x-4,\;2y+6\rangle. \]Set the gradient equal to zero:
\[ 2x-4=0, \qquad 2y+6=0. \]Hence,
\[ \boxed{(x,y)=(2,-3)}. \]What to Remember
The gradient converts a scalar field into a vector field:
\[ \boxed{ \nabla f = \langle f_x,f_y,f_z\rangle }. \]- The gradient points in the direction of steepest increase.
- The negative gradient points in the direction of steepest decrease.
- The maximum directional derivative is \(\|\nabla f\|\).
- Directional derivatives satisfy \(D_{\mathbf{u}}f=\nabla f\cdot\mathbf{u}\) for unit \(\mathbf{u}\).
- The gradient is normal to regular level curves and level surfaces.
- A field of the form \(\mathbf{F}=\nabla f\) is a gradient field.
- The scalar function \(f\) is a potential for that field.
- Critical points of differentiable scalar fields satisfy \(\nabla f=\mathbf{0}\).
Where We Go Next
The gradient showed how derivatives describe directional change in a scalar field and how a scalar field can generate a vector field.
The next lesson, Divergence and Curl, turns to two fundamental differential operations on vector fields. Divergence measures local source or sink behavior, while curl measures local rotational tendency.