Multivariable calculus

Gradients

The vector of partials: the direction of steepest increase.

• Gradient of x^2 + y^2: Vector, Steps, and GeometryThe gradient of x^2 + y^2 is the vector (2x, 2y). See both partials worked out, why it points straight away from the origin, and the mistake to avoid.
• Gradient of x^2 - y^2: The Saddle Surface Worked OutThe gradient of x^2 - y^2 is (2x, -2y). Both partials worked, why the zero gradient at the origin is a saddle and not an extremum, and the sign trap.
• Gradient of xy: Why It Is (y, x) and Not (x, y)The gradient of f(x, y) = xy is (y, x), with the variables swapped. Full method, the swap that trips students up, and the hyperbolic level curves.
• Gradient of x^3 - 3xy^2: Monkey Saddle Worked OutThe gradient of x^3 - 3xy^2 is (3x^2 - 3y^2, -6xy). Both partials in full, the mixed-term trap, and why the origin is a monkey saddle.
• Gradient of x^3 y - x y^3: Full Worked SolutionThe gradient of x^3 y - x y^3 is (3x^2 y - y^3, x^3 - 3xy^2). Every partial worked out, the exponent trap, and where the gradient vanishes.
• Gradient of x^2 + xy + y^2: Handling the Cross TermThe gradient of x^2 + xy + y^2 is (2x + y, x + 2y). See why the cross term feeds both components, and how to find the single critical point.
• Gradient of x^3 + y^3 - 3xy: Steps and Critical PointsThe gradient of x^3 + y^3 - 3xy is (3x^2 - 3y, 3y^2 - 3x). Full method, the term students drop, and how the two critical points fall out.
• Gradient of x^2 y^3: Exponent Bookkeeping Done RightThe gradient of x^2 y^3 is (2x y^3, 3x^2 y^2). Which exponent drops in which component, the product-rule trap, and where the gradient vanishes.
• Gradient of 1/(x^2+y^2+1): A Radial Bump SurfaceThe gradient of 1/(x^2+y^2+1) is (-2x, -2y) divided by (x^2+y^2+1)^2. Power-rule method, the dropped minus sign, and why it points inward.
• Gradient of x/(x^2+y^2+1): Quotient Rule WorkedThe gradient of x/(x^2+y^2+1) is ((y^2 - x^2 + 1), -2xy) over (x^2+y^2+1)^2. Quotient rule in full, the order trap, and the two critical points.
• Gradient of x^2/(y^2+1): When to Skip Quotient RuleThe gradient of x^2/(y^2+1) is (2x/(y^2+1), -2x^2 y/(y^2+1)^2). Why only one component needs the quotient rule, plus the ridge along the x-axis.
• Gradient of xy/(z^2+1): A Three Variable GradientThe gradient of xy/(z^2+1) has components y/(z^2+1), x/(z^2+1) and -2xyz/(z^2+1)^2. Which partials are easy, which needs the chain rule, and why.
• Gradient of sin(xy): Components and GeometryThe gradient of sin(xy) is the vector with components y cos(xy) and x cos(xy). See the chain rule setup, the ridges where it vanishes, and the usual slip.
• Gradient of e^x cos(y): Components and MagnitudeThe gradient of e^x cos y is the vector with components e^x cos y and minus e^x sin y. Its length is e^x at every point. Method, geometry and the sign trap.
• Gradient of ln(1 + x^2 y^2) With Full WorkingThe gradient of ln(1 + x^2 y^2) is (2xy^2, 2x^2 y) over 1 + x^2 y^2. Chain rule setup, the axes where it vanishes, and the inner derivative students drop.
• Gradient of the Gaussian Bump e^(-x^2-y^2)The gradient of e^(-x^2-y^2) is -2 e^(-x^2-y^2) times the vector (x, y). It points inward toward the peak, and is longest on the circle of radius 1 over root 2.
• Gradient of sin(x)cos(y): The Egg Carton SurfaceThe gradient of sin x cos y is (cos x cos y, -sin x sin y). Method, the grid of critical points on the egg carton surface, and the false product rule to avoid.
• Gradient of e^(xy): Components and Saddle PointThe gradient of e^(xy) is e^(xy) times the vector (y, x). See the chain rule setup, why the origin is the only critical point, and the exponent slip to avoid.
• Gradient of ln of sqrt(x^2 + y^2): The Log PotentialThe gradient of ln sqrt(x^2 + y^2) is (x, y) over x^2 + y^2. Simplify with log rules first, read off the radial field, and skip the square root chain rule.
• Gradient of arctan(y/x): The Polar Angle FieldThe gradient of arctan(y/x) is (-y, x) over x^2 + y^2. Work through the quotient inside the arctangent, and see why the field circles the origin.
• Gradient of x cos(y) + y sin(x), Term by TermThe gradient of x cos y + y sin x is (cos y + y cos x, sin x - x sin y). Work each term separately, and see why the two components are not mirror images.
• Gradient of x ln(1 + y^2) and Its Line of ZerosThe gradient of x ln(1 + y^2) is (ln(1 + y^2), 2xy over 1 + y^2). One partial needs no calculus, the other needs the chain rule, and the x axis is all critical.
• Gradient of ln(x^2 + y^2 + z^2 + 1) in Three VariablesThe gradient of ln(x^2 + y^2 + z^2 + 1) is 2(x, y, z) over x^2 + y^2 + z^2 + 1. One chain rule serves all three components of this radial field.
• Gradient of e^(xy) sin(z) in Three VariablesThe gradient of e^(xy) sin z is (y e^(xy) sin z, x e^(xy) sin z, e^(xy) cos z). See which factor is frozen for each partial and where the gradient vanishes.
• Gradient of 1/sqrt(1 + x^2 + y^2 + z^2): Inward FieldThe gradient of 1/sqrt(1 + x^2 + y^2 + z^2) is minus (x, y, z) over (1 + x^2 + y^2 + z^2)^(3/2). It points back at the origin, where the function peaks.
• Gradient of x^2 y - 3yz + z^3: Term by TermThe gradient of x^2 y - 3yz + z^3 is (2xy, x^2 - 3z, 3z^2 - 3y). The middle term feeds two components at once. Worked method and a check at (2, 1, 1).
• Gradient of x^2 y z^3: Monomial in Three VariablesThe gradient of x^2 y z^3 is (2x y z^3, x^2 z^3, 3x^2 y z^2). One exponent drops per component. Worked method, the exponent slip, and a check at (2, 3, 1).
• Gradient of e^x sin(y) cos(z): Separable ProductThe gradient of e^x sin(y) cos(z) differentiates one factor per component, giving (f, e^x cos y cos z, -e^x sin y sin z). Watch the minus sign on the z part.
• Gradient of x e^(yz): Chain Rule in the ExponentThe gradient of x e^(yz) is (e^(yz), xz e^(yz), xy e^(yz)). The y and z components pick up inner factors from the exponent. Worked check at (2, 1, 0).
• Gradient of z ln(1 + x^2 + y^2): Mixed RolesThe gradient of z ln(1 + x^2 + y^2) is (2xz/D, 2yz/D, ln D) with D = 1 + x^2 + y^2. See why the z component is the logarithm itself, not a fraction.
• Gradient of sin(x + 2y + 3z): A Fixed DirectionThe gradient of sin(x + 2y + 3z) is cos(x + 2y + 3z) times (1, 2, 3). The direction never changes, only the size and sign. Worked check at the origin.
• Gradient of x/(1 + y^2 + z^2): Quotient and ChainThe gradient of x/(1 + y^2 + z^2) is (1/D, -2xy/D^2, -2xz/D^2) with D = 1 + y^2 + z^2. One easy component, two that need the power rule.
• Gradient of xy + yz + zx: Symmetric Quadratic FormThe gradient of xy + yz + zx is (y + z, x + z, x + y): each component sums the other two variables. Includes the origin saddle and a check at (1, 2, 3).
• Gradient of arctan(xyz): Chain Rule with 1/(1+u^2)The gradient of arctan(xyz) is (yz, xz, xy) divided by 1 + x^2 y^2 z^2. It points the same way as the gradient of xyz but shrinks far from the origin.