Gradient: \nabla f = \langle f_x,\ f_y,\ f_z\rangle Directional derivative (unit vector u): D_uf = \nabla f\cdot u Unit direction from A→B: u = \frac{B-A}{\|B-A\|} Tangent plane to z=f(x,y): z - z_0 = f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0) Critical points: Solve f_x=0, f_y=0. D = f_{xx}f_{yy} - (f_{xy})^2 Min if D>0,f_{xx}>0. Max if D>0,f_{xx}<0. Saddle if D<0.

General double integral: Type I (y inner): \int_a^b \int_{g_1(x)}^{g_2(x)} f\, dy\, dx

Type II (x inner):

\int_c^d \int_{h_1(y)}^{h_2(y)} f\, dx\, dy

Polar:

x = r\cos\theta,\ y=r\sin\theta,\ dA=r\,dr\,d\theta

Circle: 0\le r\le R.

Line y=mx → \theta=\arctan m.

Region must be continuous in θ.

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3. TRIPLE INTEGRALS

Cylindrical:

x=r\cos\theta,\ y=r\sin\theta,\ dV=r\,dz\,dr\,d\theta

Spherical:

x=\rho\sin\phi\cos\theta

y=\rho\sin\phi\sin\theta

z=\rho\cos\phi,\qquad dV=\rho^2\sin\phi\, d\rho\, d\phi\, d\theta

Common surfaces:

• Sphere radius R → 0\le \rho\le R

• Cone z=\sqrt{x^2+y^2} → \phi=\pi/4

• “Above cone” → 0\le\phi\le\pi/4

• “Below cone” → \pi/4\le\phi\le\pi

• θ = full rotation unless restricted → 0\to2\pi

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4. LINE INTEGRALS

Parametric form:

\int_C \mathbf F\cdot d\mathbf r = \int_a^b \mathbf F(r(t))\cdot r'(t)\, dt

Conservative field test (2D):

P_y = Q_x

Conservative field test (3D):

\nabla\times \mathbf F = 0

If conservative:

\int_C \mathbf F\cdot d\mathbf r = f(\text{end}) - f(\text{start})

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5. GREEN, STOKES, DIVERGENCE

Green’s Theorem (2D)

Closed curve (CCW):

\oint_C P\,dx+Q\,dy = \iint_R (Q_x - P_y)\, dA

Curl:

\nabla\times F = \langle R_y-Q_z,\ P_z-R_x,\ Q_x-P_y\rangle

Divergence:

\nabla\cdot F = P_x + Q_y + R_z

Divergence Theorem (flux through closed surface):

\iint_{\partial E} F\cdot n\, dS = \iiint_E \nabla\cdot F\, dV

If divergence is constant c:

Flux = c \cdot (\text{Volume}).

Stokes’ Theorem (curve = boundary of surface):

\oint_C F\cdot d\mathbf r = \iint_S (\nabla\times F)\cdot n\, dS

Orientation:

CCW boundary ↔ upward/outward normal.

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6. SURFACE INTEGRALS

Parametrize surface r(u,v).

dS = \|\mathbf r_u \times \mathbf r_v\|\, du\, dv

Flux:

\iint_S F\cdot n\, dS = \iint F(r(u,v))\cdot (r_u\times r_v)\, du\, dv

For graphs z=g(x,y):

dS = \sqrt{1+g_x^2 + g_y^2}\, dx\, dy